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# Discrete Random Variables
## Hypergeometric Distribution (Optional)
There are five characteristics of a hypergeometric experiment:
1. You take samples from two groups.
2. You are concerned with a group of interest, called the first group.
3. You sample without replacement from the combined groups. For example, you want to choose a softball team from a combined group of 11 men and 13 women. The team consists of 10 players.
4. Each pick is not independent, since sampling is without replacement. In the softball example, the probability of picking a woman first is
. The probability of picking a man second is
if a woman was picked first. It is
if a man was picked first. The probability of the second pick depends on what happened in the first pick.
5. You are not dealing with Bernoulli trials.
The outcomes of a hypergeometric experiment fit a hypergeometric probability distribution. The random variable X = the number of items from the group of interest.
### Notation for the Hypergeometric: H = Hypergeometric Probability Distribution Function
X ~ H(r, b, n)
Read this as X is a random variable with a hypergeometric distribution. The parameters are r, b, and n: r = the size of the group of interest (first group), b = the size of the second group, n = the size of the chosen sample.
### Chapter Review
A hypergeometric experiment is a statistical experiment with the following properties:
1. You take samples from two groups
2. You are concerned with a group of interest, called the first group
3. You sample without replacement from the combined groups
4. Each pick is not independent, since sampling is without replacement
5. You are not dealing with Bernoulli trials
The outcomes of a hypergeometric experiment fit a hypergeometric probability distribution. The random variable X = the number of items from the group of interest. The distribution of X is denoted X ~ H(r, b, n), where r = the size of the group of interest (first group), b = the size of the second group, and n = the size of the chosen sample. It follows that n ≤ r + b. The mean of X is μ =
and the standard deviation is σ =
.
### Formula Review
X ~ H(r, b, n) means that the discrete random variable X has a hypergeometric probability distribution with r = the size of the group of interest (first group), b = the size of the second group, and n = the size of the chosen sample.
X = the number of items from the group of interest that are in the chosen sample, and X may take on the values x = 0, 1, . . . , up to the size of the group of interest. The minimum value for X may be larger than zero in some instances.
n ≤ r + b
The mean of X is given by the formula μ =
and the standard deviation is =
.
Use the following information to answer the next five exercises: Suppose that a group of statistics students is divided into two groups: business majors and non-business majors. There are 16 business majors in the group and seven non-business majors in the group. A random sample of nine students is taken. We are interested in the number of business majors in the sample.
### HOMEWORK
|
# Discrete Random Variables
## Poisson Distribution (Optional)
There are two main characteristics of a Poisson experiment.
1. The Poisson probability distribution gives the probability of a number of events occurring in a fixed interval of time or space if these events happen with a known average rate and independently of the time since the last event. For example, a book editor might be interested in the number of words spelled incorrectly in a particular book. It might be that, on the average, there are five words spelled incorrectly in 100 pages. The interval is the 100 pages.
2. The Poisson distribution may be used to approximate the binomial if the probability of success is small (such as .01) and the number of trials is large (such as 1,000). You will verify the relationship in the homework exercises. n is the number of trials, and p is the probability of a success.
The random variable X = the number of occurrences in the interval of interest.
### Notation for the Poisson: P = Poisson Probability Distribution Function
X ~ P(μ)
Read this as X is a random variable with a Poisson distribution. The parameter is μ (or λ); μ (or λ) = the mean for the interval of interest.
### References
Centers for Disease Control and Prevention. (2012, Oct. 2). Teen drivers: Get the facts. Retrieved from http://www.cdc.gov/Motorvehiclesafety/Teen_Drivers/teendrivers_factsheet.html
Daily Mail. (2011, June 9). One born every minute: the maternity unit where mothers are THREE to a bed. Retrieved from http://www.dailymail.co.uk/news/article-2001422/Busiest-maternity-ward-planet-averages-60-babies-day-mothersbed.html
Department of Aviation at the Hartsfield-Jackson Atlanta International Airport. (2013). ATL fact sheet. Retrieved from http://www.atlanta-airport.com/Airport/ATL/ATL_FactSheet.aspx
Lenhart, A. (2012). Teens, smartphones & testing: Texting volume is up while the frequency of voice calling is down. About one in four teens say they own smartphones. Pew Internet. Retrieved from http://www.pewinternet.org/~/media/Files/Reports/2012/PIP_Teens_Smartphones_and_Texting.pdf
Ministry of Health, Labour, and Welfare. (n.d.). Children and childrearing. Retrieved from http://www.mhlw.go.jp/english/policy/children/children-childrearing/index.html
Pew Internet. (2013). How Americans use text messaging. Retrieved from http://pewinternet.org/Reports/2011/Cell-Phone-Texting-2011/Main-Report.aspx
South Carolina Department of Mental Health. (2006). Eating disorder statistics. Retrieved from http://www.state.sc.us/dmh/anorexia/statistics.htm
The Guardian. (2011, June 8). Giving birth in Manila: The maternity ward at the Dr Jose Fabella Memorial Hospital in Manila, the busiest in the Philippines, where there is an average of 60 births a day. Retrieved from http://www.theguardian.com/world/gallery/2011/jun/08/philippines-health#/?picture=375471900&index=2
Vanderkam, L. (2012, Oct. 8). Stop checking your email, now. CNNMoney. Retrieved from http://management.fortune.cnn.com/2012/10/08/stop-checking-your-email-now/
World Earthquakes. (2012). World earthquakes: Live earthquake news and highlights. Retrieved from http://www.worldearthquakes.com/index.php?option=ethq_prediction
### Chapter Review
A Poisson probability distribution of a discrete random variable gives the probability of a number of events occurring in a fixed interval of time or space, if these events happen at a known average rate and independently of the time since the last event. The Poisson distribution may be used to approximate the binomial, if the probability of success is small (less than or equal to .05) and the number of trials is large (greater than or equal to 20).
### Formula Review
X ~ P(μ) means that X has a Poisson probability distribution where X = the number of occurrences in the interval of interest.
X takes on the values x = 0, 1, 2, 3, . . .
The mean μ is typically given.
The variance is σ2 = μ, and the standard deviation is
.
When P(μ) is used to approximate a binomial distribution, μ = np where n represents the number of independent trials and p represents the probability of success in a single trial.
Use the following information to answer the next six exercises: On average, a clothing store gets 120 customers per day.
Use the following information to answer the next six exercises: On average, eight teens in the United States die from motor vehicle injuries per day. As a result, states across the country are debating raising the driving age.
### HOMEWORK
Use the following information to answer the next two exercises: The average number of times per week that Mrs. Plum’s cats wake her up at night because they want to play is 10. We are interested in the number of times her cats wake her up each week. |
# Continuous Random Variables
## Introduction
Continuous random variables have many applications. Baseball batting averages, IQ scores, the length of time a long-distance telephone call lasts, the amount of money a person carries, the length of time a computer chip lasts, and SAT scores are just a few. The field of reliability depends on a variety of continuous random variables.
### Properties of Continuous Probability Distributions
The graph of a continuous probability distribution is a curve. Probability is represented by the area under the curve.
The curve is called the probability density function (abbreviated as pdf). We use the symbol f(x) to represent the curve. f(x) is the function that corresponds to the graph; we use the density function f(x) to draw the graph of the probability distribution.
Area under the curve is given by a different function called the cumulative distribution function (abbreviated as cdf). The cumulative distribution function is used to evaluate probability as area.
We will find the area that represents probability by using geometry, formulas, technology, or probability tables. In general, calculus is needed to find the area under the curve for many probability density functions. When we use formulas to find the area in this textbook, we are using formulas that were found by using the techniques of integral calculus. However, because most students taking this course have not studied calculus, we will not be using calculus in this textbook.
There are many continuous probability distributions. When probability is modeled by use of a continuous probability distribution, the distribution used is selected to model and fit the particular situation in the best way.
In this chapter and the next, we will study the uniform distribution, the exponential distribution, and the normal distribution. The following graphs illustrate these distributions:
|
# Continuous Random Variables
## Continuous Probability Functions
We begin by defining a continuous probability density function. We use the function notation f(x). Intermediate algebra may have been your first formal introduction to functions. In the study of probability, the functions we study are special. We define the function f(x) so that the area between it and the x-axis is equal to a probability. Since the maximum probability is one, the maximum area is also one. For continuous probability distributions, PROBABILITY = AREA.
### Chapter Review
The probability density function (pdf) is used to describe probabilities for continuous random variables. The area under the density curve between two points corresponds to the probability that the variable falls between those two values. In other words, the area under the density curve between points a and b is equal to P(a < x < b). The cumulative distribution function (cdf) gives the probability as an area. If X is a continuous random variable, the probability density function (pdf), f(x), is used to draw the graph of the probability distribution. The total area under the graph of f(x) is one. The area under the graph of f(x) and between values a and b gives the probability P(a < x < b).
The cumulative distribution function (cdf) of X is defined by P (X ≤ x). It is a function of x that gives the probability that the random variable is less than or equal to x.
### Formula Review
Probability density function (pdf) f(x):
1. f(x) ≥ 0
2. The total area under the curve f(x) is one.
Cumulative distribution function (cdf): P(X ≤ x)
### Homework
For each probability and percentile problem, draw the picture. |
# Continuous Random Variables
## The Uniform Distribution
The uniform distribution is a continuous probability distribution and is concerned with events that are equally likely to occur. When working out problems that have a uniform distribution, be careful to note if the data are inclusive or exclusive of endpoints.
### Chapter Review
If X has a uniform distribution where a < x < b or a ≤ x ≤ b, then X takes on values between a and b (may include a and b). All values x are equally likely. We write X ∼ U(a, b). The mean of X is
. The standard deviation of X is
. The probability density function of X is
for a ≤ x ≤ b. The cumulative distribution function of X is P(X ≤ x) =
. X is continuous.
The probability P(c < X < d) may be found by computing the area under f(x), between c and d. Since the corresponding area is a rectangle, the area may be found simply by multiplying the width and the height.
### Formula Review
X = a real number between a and b (in some instances, X can take on the values a and b). a = smallest X, b = largest X
X ~ U(a, b)
The mean is
The standard deviation is
Probability density function:
for
Area to the left of P(X < x) = (x – a)
Area to the right of P(X > x) = (b – x)
Area between P(c < x < d) = (base)(height) = (d – c)
Uniform: X ~ U(a, b) where a < x < b
1. pdf:
for a ≤ x ≤ b
2. cdf: P(X ≤ x) =
3. mean µ =
4. standard deviation σ
5. P(c < X < d) = (d – c)
### References
McDougall, J. A. (1995). The McDougall program for maximum weight loss. New York: Plume
Use the following information to answer the next 10 questions. The data that follow are the square footage (in 1,000 feet squared) of 28 homes:
The sample mean = 2.50 and the sample standard deviation = 0.8302.
The distribution can be written as X ~ U(1.5, 4.5).
Use the following information to answer the next eight exercises. A distribution is given as X ~ U(0, 12).
Use the following information to answer the next 12 exercises. The age of cars in the staff parking lot of a suburban college is uniformly distributed from six months (0.5 years) to 9.5 years.
### Homework
For each probability and percentile problem, draw the picture.
Use the following information to answer the next three exercises. The Sky Train from the terminal to the rental–car and long–term parking center is supposed to arrive every eight minutes. The waiting times for the train are known to follow a uniform distribution. |
# Continuous Random Variables
## The Exponential Distribution (Optional)
The exponential distribution is often concerned with the amount of time until some specific event occurs. For example, the amount of time (beginning now) until an earthquake occurs has an exponential distribution. Other examples include the length, in minutes, of long-distance business telephone calls, and the amount of time, in months, a car battery lasts. It can be shown, too, that the value of the change that you have in your pocket or purse approximately follows an exponential distribution.
Values for an exponential random variable occur in the following way. There are fewer large values and more small values. For example, the amount of money customers spend in one trip to the supermarket follows an exponential distribution. There are more people who spend small amounts of money and fewer people who spend large amounts of money.
Exponential distributions are commonly used in calculations of product reliability, or the length of time a product lasts.
### Memorylessness of the Exponential Distribution
In recall that the amount of time between customers is exponentially distributed with a mean of two minutes (X ~ Exp(0.5)). Suppose that five minutes have elapsed since the last customer arrived. Since an unusually long amount of time has now elapsed, it would seem to be more likely for a customer to arrive within the next minute. With the exponential distribution, this is not the case—the additional time spent waiting for the next customer does not depend on how much time has already elapsed since the last customer. This is referred to as the memoryless property. Specifically, the memoryless property says the following
For example, if five minutes have elapsed since the last customer arrived, then the probability that more than one minute will elapse before the next customer arrives is computed by using r = 5 and t = 1 in the foregoing equation.
This is the same probability as that of waiting more than one minute for a customer to arrive after the previous arrival.
The exponential distribution is often used to model the longevity of an electrical or a mechanical device. In , the lifetime of a certain computer part has the exponential distribution with a mean of ten years (X ~ Exp(0.1)). The memoryless property says that knowledge of what has occurred in the past has no effect on future probabilities. In this case it means that an old part is not any more likely to break down at any particular time than a brand new part. In other words, the part stays as good as new until it suddenly breaks. For example, if the part has already lasted ten years, then the probability that it lasts another seven years is P(X > 17|X > 10) = P(X > 7) = 0.4966.
### Relationship Between the Poisson and the Exponential Distribution
There is an interesting relationship between the exponential distribution and the Poisson distribution. Suppose that the time that elapses between two successive events follows the exponential distribution with a mean of μ units of time. Also assume that these times are independent, meaning that the time between events is not affected by the times between previous events. If these assumptions hold, then the number of events per unit time follows a Poisson distribution with mean λ = 1/μ. Recall from the chapter on Discrete Random Variables that if X has the Poisson distribution with mean λ, then
. Conversely, if the number of events per unit time follows a Poisson distribution, then the amount of time between events follows the exponential distribution. (k! = k*(k–1*)(k–2)*(k–3)…3*2*1)
### Chapter Review
If X has an exponential distribution with mean μ, then the decay parameter is m =
, and we write X ~ Exp(m) where x ≥ 0 and m > 0 . The probability density function of X is f(x) = me (or equivalently
. The cumulative distribution function of X is P(X ≤ x) = 1 – e–.
The exponential distribution has the memoryless property, which says that future probabilities do not depend on any past information. Mathematically, it says that P(X > x + k|X > x) = P(X > k).
If T represents the waiting time between events, and if T ~ Exp(λ), then the number of events X per unit time follows the Poisson distribution with mean λ. The probability density function of X is
. This may be computed using a TI-83, 83+, 84, 84+ calculator with the command poissonpdf(λ, k). The cumulative distribution function P(X ≤ k) may be computed using the TI-83, 83+,84, 84+ calculator with the command poissoncdf(λ, k).
### Formula Review
Exponential: X ~ Exp(m) where m = the decay parameter
1. pdf: f(x) = me(– where x ≥ 0 and m > 0
2. cdf: P(X ≤ x) = 1 – e(–
3. mean µ =
4. standard deviation σ = µ
5. percentile k: k =
6. Additionally
7. Memoryless property: P(X > x + k|X > x) = P (X > k)
8. Poisson probability:
with mean λ
9. k! = k*(k−1)*(k−2)*(k−3)*…3*2*1
### References
Baseball-Reference.com. (2013). No-hitter. Retrieved from http://www.baseball-reference.com/bullpen/No-hitter
U.S. Census Bureau. (n.d.). Retrieved from https://www.census.gov/
World Earthquakes. (2013). Earthquake data for Papua New Guinea. Retrieved from http://www.world-earthquakes.com/
Zhou, Rick. (2013). Exponential distribution lecture slides. Retrieved from www.public.iastate.edu/~riczw/stat330s11/lecture/lec13.pdf
Use the following information to answer the next 10 exercises. A customer service representative must spend different amounts of time with each customer to resolve various concerns. The amount of time spent with each customer can be modeled by the following distribution: X ~ Exp(0.2)
Use the following information to answer the next eight exercises. A distribution is given as X ~ Exp(0.75).
Use the following information to answer the next eight exercises. Carbon-14 is a radioactive element with a half-life of about 5,730 years. Carbon-14 is said to decay exponentially. The decay rate is 0.000121. We start with one gram of carbon-14. We are interested in the time (years) it takes to decay carbon-14.
### Homework
Use the following information to answer the next three exercises. The average lifetime of a certain new cell phone is three years. The manufacturer will replace any cell phone failing within two years of the date of purchase. The lifetime of these cell phones is known to follow an exponential distribution. |
# The Normal Distribution
## Introduction
The normal, a continuous distribution, is the most important of all the distributions. It is widely used and even more widely abused. Its graph is bell-shaped. You see the bell curve in almost all disciplines, including psychology, business, economics, the sciences, nursing, and, of course, mathematics. Some of your instructors may use the normal distribution to help determine your grade. Most IQ scores are normally distributed. Often, real-estate prices fit a normal distribution. The normal distribution is extremely important, but it cannot be applied to everything in the real world.
In this chapter, you will study the normal distribution, the standard normal distribution, and applications associated with them.
The normal distribution has two parameters: the mean (μ) and the standard deviation (σ). If X is a quantity to be measured that has a normal distribution with mean (μ) and standard deviation (σ), we designate this by writing
The curve is symmetric about a vertical line drawn through the mean, μ. In theory, the mean is the same as the median, because the graph is symmetric about μ. With a normal distribution, the mean, median, and mode all lie at the same point. The normal distribution depends only on the mean and the standard deviation. The location of the mean simply indicates the location of the line of symmetry, in a normal distribution. Since the area under the curve must equal one, a change in the standard deviation, σ, causes a change in the shape of the curve; the curve becomes fatter or skinnier depending on σ. A change in μ causes the graph to shift to the left or right. The location of the mean simply indicates the location of the line of symmetry, in a normal distribution. This means there are an infinite number of normal probability distributions. One distribution of special interest is called the standard normal distribution.
### Formula Review
X ∼ N(μ, σ)
μ = the mean, σ = the standard deviation |
# The Normal Distribution
## The Standard Normal Distribution
The standardized normal distribution is a type of normal distribution, with a mean of 0 and standard deviation of 1. It represents a distribution of standardized scores, called , as opposed to raw scores (the actual data values). A indicates the number of standard deviation a score falls above or below the mean. Z-scores allow for comparison of scores, occurring in different data sets, with different means and standard deviations. It would not make sense to compare apples and oranges. Likewise, it does not make sense to compare scores from two different samples that have different means and standard deviations. Z-scores can be looked up in a Z-Table of Standard Normal Distribution, in order to find the area under the standard normal curve, between a score and the mean, between two scores, or above or below a score. The standard normal distribution allows us to interpret standardized scores and provides us with one table that we may use, in order to compute areas under the normal curve, for an infinite number of data sets, no matter what the mean or standard deviation.
A z-score is calculated as
. The score itself can be found by using algebra and solving for x. Multiplying both sides of the equation by σ gives:
. Adding μ to both sides of the equation gives
.
Suppose we have a data set with a mean of 5 and standard deviation of 2. We want to determine the number of standard deviations the score of 11 falls above the mean. We can find this answer (or z-score) by writing
or
we can solve for z.
We have determined that the score of 11 falls 3 standard deviations above the mean of 5.
With a standard normal distribution, we indicate the distribution by writing Z ~ N(0, 1) which shows the normal distribution has a mean of 0 and standard deviation of 1. This notation simply indicates that a standard normal distribution is being used.
### Z-Scores
As described previously, if X is a normally distributed random variable and X ~ N(μ, σ), then the z-score is
The z-score tells you how many standard deviations the value x is above, to the right of, or below, to the left of, the mean, μ. Values of x that are larger than the mean have positive z-scores, and values of x that are smaller than the mean have negative z-scores. If x equals the mean, then x has a z-score of zero.
When determining the z-score for an x-value, for a normal distribution, with a given mean and standard deviation, the notation above for a normal distribution, will be given.
The Empirical Rule If X is a random variable and has a normal distribution with mean µ and standard deviation σ, then the Empirical Rule states the following:
1. About 68 percent of the x values lie between –1σ and +1σ of the mean µ (within one standard deviation of the mean).
2. About 95 percent of the x values lie between –2σ and +2σ of the mean µ (within two standard deviations of the mean).
3. About 99.7 percent of the x values lie between –3σ and +3σ of the mean µ (within three standard deviations of the mean). Notice that almost all the x values lie within three standard deviations of the mean.
4. The z-scores for +1σ and –1σ are +1 and –1, respectively.
5. The z-scores for +2σ and –2σ are +2 and –2, respectively.
6. The z-scores for +3σ and –3σ are +3 and –3, respectively.
So, in other words, this is that about 68 percent of the values lie between z-scores of –1 and 1, about 95% of the values lie between z-scores of –2 and 2, and about 99.7 percent of the values lie between z-scores of -3 and 3. These facts can be checked, by looking up the mean to z area in a z-table for each positive z-score and multiplying by 2.
The empirical rule is also known as the 68–95–99.7 rule.
### References
CollegeBoard. (2012). 2012 College-bound seniors: Total group profile report. Retrieved from http://media.collegeboard.com/digitalServices/pdf/research/TotalGroup-2012.pdf
Joyce, C. A., Janssen, S., & Liu, M. L. (2010). The world almanac and book of facts, 2010. New York, NY: World Almanac Books.
London School of Hygiene and Tropical Medicine. (2009). Calculation of z-scores. Retrieved from http://conflict.lshtm.ac.uk/page_125.htm
National Center for Education Statistics. (2009). ACT score averages and standard deviations, by sex and race/ethnicity, and percentage of ACT test takers, by selected composite score ranges and planned fields of study: Selected years, 1995 through 2009. Retrieved from http://nces.ed.gov/programs/digest/d09/tables/dt09_147.asp
NBA.com. (2013). NBA Media Ventures. Retrieved from http://www.nba.com
StatCrunch. (2010). Blood pressure of males and females. Retrieved from http://www.statcrunch.com/5.0/viewreport.php?reportid=11960
The Mercury News. (n.d.). Retrieved from http://www.mercurynews.com/
Wikipedia. (2013). List of stadiums by capacity - Wikipedia. Retrieved from https://en.wikipedia.org/wiki/List_of_stadiums_by_capacity
### Chapter Review
A z-score is a standardized value. Its distribution is the standard normal, Z ~ N(0, 1). The mean of the z-scores is zero and the standard deviation is one. If z is the z-score for a value x from the normal distribution N(µ, σ), then z tells you how many standard deviations x is abovegreater thanor belowless thanµ.
### Formula Review
Z ~ N(0, 1)
z = a standardized value (z-score)
mean = 0, standard deviation = 1
To find the kth percentile of X when the z-score is known,k = μ + (z)σ
z-score: z =
Z = the random variable for z-scores
Use the following information to answer the next two exercises: The life of Sunshine CD players is normally distributed with mean of 4.1 years and a standard deviation of 1.3 years. A CD player is guaranteed for three years. We are interested in the length of time a CD player lasts.
### Homework
Use the following information to answer the next two exercises: The patient recovery time from a particular surgical procedure is normally distributed with a mean of 5.3 days and a standard deviation of 2.1 days. |
# The Normal Distribution
## Using the Normal Distribution
The shaded area in the following graph indicates the area to the left of x. This area could represent the percentage of students scoring less than a particular grade on a final exam. This area is represented by the probability P(X < x). Normal tables, computers, and calculators are used to provide or calculate the probability P(X < x).
The area to the right is then P(X > x) = 1 – P(X < x). Remember, P(X < x) = Area to the left of the vertical line through x. P(X < x) = 1 – P(X < x) = Area to the right of the vertical line through x. P(X < x) is the same as P(X ≤ x) and P(X > x) is the same as P(X ≥ x) for continuous distributions.
Suppose the graph above were to represent the percentage of students scoring less than 75 on a final exam, with this probability equal to 0.39. This would also indicate that the percentage of students scoring higher than 75 was equal to 1 minus 0.39 or 0.61.
### Calculations of Probabilities
Probabilities are calculated using technology. There are instructions given as necessary for the TI-83+ and TI-84 calculators.
### References
Chicago Public Media & Ira Glass. (2013). 403: NUMMI. Retrieved from http://www.thisamericanlife.org/radioarchives/episode/403/nummi
lauramitchell347. (2012, Dec. 28). Smart phone users, by the numbers. Visually. Retrieved from http://visual.ly/smart-phone-users-numbers
Statistics Brain Research Institute. (2013). Facebook company statistics statistic brain. Retrieved from http://www.statisticbrain.com/facebook-statistics/
Wikipedia (2013). Naegele's rule. Retrieved from http://en.wikipedia.org/wiki/Naegele's_rule
Win at the Lottery. (2013). Scratch-off lottery ticket playing tips. Retrieved from www.winatthelottery.com/public/department40.cfm
### Chapter Review
The normal distribution, which is continuous, is the most important of all the probability distributions. Its graph is bell-shaped. This bell-shaped curve is used in almost all disciplines. Since it is a continuous distribution, the total area under the curve is one. The parameters of the normal are the mean µ and the standard deviation σ. A special normal distribution, called the standard normal distribution, is the distribution of z-scores. Its mean is zero, and its standard deviation is one.
### Formula Review
Normal Distribution: X ~ N(µ, σ), where µ is the mean and σ is the standard deviation
Standard Normal Distribution: Z ~ N(0, 1).
Calculator function for probability: normalcdf (lower x value of the area, upper x value of the area, mean, standard deviation)
Calculator function for the kth percentile: k = invNorm (area to the left of k, mean, standard deviation)
Use the following information to answer the next four exercises:
X ~ N(54, 8)
### Homework
Use the following information to answer the next two exercises: The patient recovery time from a particular surgical procedure is normally distributed with a mean of 5.3 days and a standard deviation of 2.1 days.
Use the following information to answer the next three exercises: The length of time it takes to find a parking space at 9 a.m. follows a normal distribution with a mean of five minutes and a standard deviation of two minutes. |
# The Central Limit Theorem
## Introduction
Why are we so concerned with means? Two reasons are they give us a middle ground for comparison, and they are easy to calculate. In this chapter, you will study means and the central limit theorem.
The central limit theorem (clt) is one of the most powerful and useful ideas in all of statistics. There are two alternative forms of the theorem, and both alternatives are concerned with drawing a finite samples size n from a population with a known mean, μ, and a known standard deviation, σ. The first alternative says that if we collect samples of size n with a large enough n, calculate each sample's mean, and create a histogram of those means, then the resulting histogram will tend to have an approximate normal bell shape. The second alternative says that if we again collect samples of size n that are large enough, calculate the sum of each sample and create a histogram, then the resulting histogram will again tend to have a normal bell shape. The central limit theorem for sample means is more discussed in the world of statistics, but it is important to note that taking each sample's sum and graphing the sums will also result in a normal histogram. There are instances where one wishes to calculate the sum of a sample, as opposed to its mean.
In either case, it does not matter what the distribution of the original population is, or whether you even need to know it. The important fact is that the distributions of sample means and the sums tend to follow the normal distribution.
The size of the sample, n, that is required in order to be large enough depends on the original population from which the samples are drawn (the sample size should be at least 30 or the data should come from a normal distribution). If the original population is far from normal, then more observations are needed for the sample means or sums to be normal. Sampling is done with replacement. |
# The Central Limit Theorem
## The Central Limit Theorem for Sample Means (Averages)
Suppose X is a random variable with a distribution that may be known or unknown (it can be any distribution). Using a subscript that matches the random variable, suppose
1.
= the mean of X
2.
= the standard deviation of X
If you draw random samples of size n, then as n increases, the random variable
, which consists of sample means, tends to be normally distributed and
The central limit theorem for sample means says that if you keep drawing larger and larger samples (such as rolling one, two, five, and finally, ten dice) and calculating their means, the sample means form their own normal distribution (the sampling distribution). The normal distribution has the same mean as the original distribution and a variance that equals the original variance divided by the sample size. The variable n is the number of values that are averaged together, not the number of times the experiment is done.
To put it more formally, if you draw random samples of size n, the distribution of the random variable
, which consists of sample means, is called the sampling distribution of the mean. The sampling distribution of the mean approaches a normal distribution as n, the sample size, increases.
The random variable
has a different z-score associated with it from that of the random variable X. The mean
is the value of
in one sample.
μ is the average of both X and
.
= standard deviation of
and is called the standard error of the mean.
### References
Baran, D. (2010). 20 percent of Americans have never used email. WebGuild. Retrieved from http://www.webguild.org/20080519/20-percent-of-americans-have-never-used-email
The Flurry Blog. (2013). Retrieved from http://blog.flurry.com
U.S. Department of Agriculture. (n.d.). Retrieved from https://www.usda.gov/
### Chapter Review
In a population whose distribution may be known or unknown, if the size (n) of the sample is sufficiently large, the distribution of the sample means will be approximately normal. The mean of the sample means will equal the population mean. The standard deviation of the distribution of the sample means, called the standard error of the mean, is equal to the population standard deviation divided by the square root of the sample size (n).
### Formula Review
Central limit theorem for sample means:
~ N
Mean
: μ
Central limit theorem for sample means z-score and standard error of the mean:
Standard error of the mean (standard deviation (
)):
Use the following information to answer the next six exercises: Yoonie is a personnel manager in a large corporation. Each month she must review 16 of the employees. From past experience, she has found that the reviews take her approximately four hours each to do with a population standard deviation of 1.2 hours. Let Χ be the random variable representing the time it takes her to complete one review. Assume Χ is normally distributed. Let
be the random variable representing the mean time to complete the 16 reviews. Assume that the 16 reviews represent a random set of reviews.
### Homework
|
# The Central Limit Theorem
## The Central Limit Theorem for Sums (Optional)
Suppose X is a random variable with a distribution that may be known or unknown (it can be any distribution) and suppose:
1. μ = the mean of Χ
2. σ = the standard deviation of X
If you draw random samples of size n, then as n increases, the random variable ΣX consisting of sums tends to be normally distributed and ΣΧ ~ N[(n)(μ), (
)(σ)].
The central limit theorem for sums says that if you keep drawing larger and larger samples and taking their sums, the sums form their own normal distribution (the sampling distribution), which approaches a normal distribution as the sample size increases. The normal distribution has a mean equal to the original mean multiplied by the sample size and a standard deviation equal to the original standard deviation multiplied by the square root of the sample size.
The random variable ΣX has the following z-score associated with it:
1. Σx is one sum.
2.
### References
Farago, P. (2012, Oct. 29). The truth about cats and dogs: Smartphone vs tablet usage differences. Flurry Analytics Blog. Retrieved from http://flurrymobile.tumblr.com/post/113379683050/the-truth-about-cats-and-dogs-smartphone-vs
### Chapter Review
The central limit theorem tells us that for a population with any distribution, the distribution of the sums for the sample means approaches a normal distribution as the sample size increases. In other words, if the sample size is large enough, the distribution of the sums can be approximated by a normal distribution, even if the original population is not normally distributed. Additionally, if the original population has a mean of μ and a standard deviation of σ, the mean of the sums is nμ and the standard deviation is
(σ), where n is the sample size.
### Formula Review
Central limit theorem for sums: ∑X ~ N[(n)(μ),(
)(σ)]
Mean for sums (∑X): (n)(μ)
Central limit theorem for sums z-score and standard deviation for sums:
Standard deviation for sums (∑X):
(σ)
Use the following information to answer the next four exercises: An unknown distribution has a mean of 80 and a standard deviation of 12. A sample size of 95 is drawn randomly from the population.
Use the following information to answer the next five exercises: The distribution of results from a cholesterol test has a mean of 180 and a standard deviation of 20. A sample size of 40 is drawn randomly.
Use the following information to answer the next six exercises: A researcher measures the amount of sugar in several cans of the same type of soda. The mean is 39.01 with a standard deviation of 0.5. The researcher randomly selects a sample of 100.
Use the following information to answer the next four exercises: An unknown distribution has a mean 12 and a standard deviation of one. A sample size of 25 is taken. Let X = the object of interest.
A market researcher analyzes how many electronics devices customers buy in a single purchase. The distribution has a mean of three with a standard deviation of 0.7. She samples 400 customers.
An unkwon distribution has a mean of 100, a standard deviation of 100, and a sample size of 100. Let X = one object of interest.
### Homework
|
# The Central Limit Theorem
## Using the Central Limit Theorem
It is important for you to understand when to use the central limit theorem. If you are being asked to find the probability of the mean, use the clt for the means. If you are being asked to find the probability of a sum or total, use the clt for sums. This also applies to percentiles for means and sums.
### Examples of the Central Limit Theorem
### Law of Large Numbers
The law of large numbers says that if you take samples of larger and larger sizes from any population, then the mean
of the samples tends to get closer and closer to μ. From the central limit theorem, we know that as n gets larger and larger, the sample means follow a normal distribution. The larger n gets, the smaller the standard deviation gets. (Remember that the standard deviation for
is
.) This means that the sample mean
must be close to the population mean μ. We can say that μ is the value that the sample means approach as n gets larger. The central limit theorem illustrates the law of large numbers.
### Central Limit Theorem for the Mean and Sum Examples
### References
The Wall Street Journal. (n.d.). Retrieved from https://www.wsj.com/
Centers for Disease Control and Prevention. (2017, April 16). National health and nutrition examination survey. National Center for Health Statistics. Retrieved from http://www.cdc.gov/nchs/nhanes.htm
### Chapter Review
The central limit theorem can be used to illustrate the law of large numbers. The law of large numbers states that the larger the sample size you take from a population, the closer the sample mean,
, gets to μ.
Use the following information to answer the next 10 exercises: A manufacturer produces 25-pound lifting weights. The lowest actual weight is 24 pounds, and the highest is 26 pounds. Each weight is equally likely, so the distribution of weights is uniform. A sample of 100 weights is taken.
The length of time a particular smartphone's battery lasts follows an exponential distribution with a mean of ten months. A sample of 64 of these smartphones is taken.
A uniform distribution has a minimum of six and a maximum of ten. A sample of 50 is taken.
### Homework
Richard’s Furniture Company delivers furniture from 10 a.m. to 2 p.m. continuously and uniformly. We are interested in how long (in hours) past the 10 a.m. start time that individuals wait for their delivery.
Use the following information to answer the next two exercises: The time to wait for a particular rural bus is distributed uniformly from zero to 75 minutes. One hundred riders are randomly sampled to learn how long they waited.
The cost of unleaded gasoline in the Bay Area once followed an unknown distribution with a mean of $4.59 and a standard deviation of $0.10. Sixteen gas stations from the Bay Area are randomly chosen. We are interested in the average cost of gasoline for the 16 gas stations.
For each problem, wherever possible, provide graphs and use a calculator. |
# Confidence Intervals
## Introduction
Suppose you were trying to determine the mean rent of a two-bedroom apartment in your town. You might look in the classified section of the newspaper, write down several rents listed, and average them together. You would have obtained a point estimate of the true mean. If you are trying to determine the percentage of times you make a basket when shooting a basketball, you might count the number of shots you make and divide that by the number of shots you attempt. In this case, you would have obtained a point estimate for the true proportion.
We use sample data to make generalizations about an unknown population. This part of statistics is called inferential statistics. The sample data help us to make an estimate of a population . We realize that the point estimate is most likely not the exact value of the population parameter, but close to it. After calculating point estimates, we construct interval estimates, called confidence intervals.
In this chapter, you will learn to construct and interpret confidence intervals. You will also learn a new distribution, the Student's-t, and how it is used with those intervals. Throughout the chapter, it is important to keep in mind that the confidence interval is a random variable. It is the population parameter that is fixed.
If you worked in the marketing department of an entertainment company, you might be interested in the mean number of songs a consumer downloads a month from an internet music store. If so, you could conduct a survey and calculate the sample mean,
and the sample standard deviation, s. You would use
to estimate the population mean and s to estimate the population standard deviation. The sample mean,
, is the point estimate for the population mean, μ. The sample standard deviation, s, is the point estimate for the population standard deviation, σ.
Each instance of
and s is called a statistic.
A confidence interval is another type of estimate but, instead of being just one number, it is an interval of numbers. The interval of numbers is a range of values calculated from a given set of sample data. The confidence interval is likely to include an unknown population parameter.
Suppose, for the internet music example, we do not know the population mean, μ, but we do know that the population standard deviation is σ = 1 and our sample size is 100. Then, by the central limit theorem, the standard deviation for the sample mean is
The Empirical Rule, which applies to bell-shaped distributions, says that in approximately 95 percent of the samples, the sample mean,
, will be within two standard deviations of the population mean, μ. For our internet music example, two standard deviations would be calculated as (2)(0.1) = 0.2. The sample mean,
is likely to be within 0.2 units of μ.
In this example, we do not know the true population mean μ (because we do not have information from all the internet music users!), but we can compute the sample mean
based on our sample of 100 individuals. Because the sample mean is likely to be within 0.2 units of the true population mean 95 percent of the times that we take a sample of 100 users, we can say with 95 percent confidence that μ is within 0.2 units of
. In other words, μ is somewhere between
and
.
Suppose that from the sample of 100 internet music customers, we compute a sample mean download of
songs per month. Since we know that the population standard deviation is
, according to the central limit theorem, the standard deviation for the sample means is
.
We know that there is a 95 percent chance that the true population mean value μ is between two standard deviations from the sample mean. That is, with 95 percent confidence we can say that μ is between
and
.
Replacing the symbols for their values in this example, we say that we are 95 percent confident that the true average number of songs downloaded from an internet music store per month is between
The 95 percent confidence interval for μ is (1.8, 2.2).
The 95 percent confidence interval implies two possibilities. Either the interval (1.8, 2.2) contains the true mean, μ, or our sample produced an
that is not within 0.2 units of the true mean μ. The second possibility happens for only 5 percent of all the samples (95–100 percent).
Remember that a confidence interval is created for an unknown population parameter like the population mean, μ. Confidence intervals for some parameters have the form
(point estimate – margin of error, point estimate + margin of error).
The margin of error depends on the confidence level or percentage of confidence and the standard error of the mean.
When you read newspapers and journals, you might notice that some reports use the phrase margin of error. Other reports will not use that phrase, but include a confidence interval as the point estimate plus or minus the margin of error. Those are two ways of expressing the same concept. |
# Confidence Intervals
## A Single Population Mean Using the Normal Distribution
A confidence interval for a population mean with a known standard deviation is based on the fact that the sample means follow an approximately normal distribution. Suppose that our sample has a mean of
and we have constructed the 90 percent confidence interval (5, 15), where the margin of error = 5.
### Calculating the Confidence Interval
To construct a confidence interval for a single unknown population mean, μ, where the population standard deviation is known, we need
as an estimate for μ, and we need the margin of error. The margin of error for the population mean is called the error bound for a population mean (EBM). The sample mean,
is the point estimate of the unknown population mean, μ.
The confidence interval (CI) estimate will have the form:
(point estimate – error bound, point estimate + error bound) or, in symbols, (
).
The margin of error (EBM) depends on the confidence level (). The confidence level is often considered the probability that the calculated confidence interval estimate will contain the true population parameter. However, it is more accurate to state that the confidence level is the percentage of confidence intervals that contain the true population parameter when repeated samples are taken. Most often, the person constructing the confidence interval will choose a confidence level of 90 percent or higher, because that person wants to be reasonably certain of his or her conclusions.
Another probability, which is called alpha
is related to the confidence level, CL. Alpha is the probability that the confidence interval does not contain the unknown population parameter. Mathematically, alpha can be computed as
.
A confidence interval for a population mean with a known standard deviation is based on the fact that the sample means follow an approximately normal distribution. Suppose that our sample has a mean of
= 10, and we have constructed the 90 percent confidence interval (5, 15) where EBM = 5.
To get a 90 percent confidence interval, we must include the central 90 percent of the probability of the normal distribution. If we include the central 90 percent, we leave out a total of α = 10 percent in both tails, or 5 percent in each tail, of the normal distribution.
The critical value 1.645 is the in a standard normal probability distribution that puts an area of 0.90 in the center, an area of 0.05 in the far left tail, and an area of 0.05 in the far right tail. To capture the central 90 percent, we must go out 1.645 standard deviations on either side of the calculated sample mean. The critical value will change depending on the confidence level of the interval.
It is important that the standard deviation used be appropriate for the parameter we are estimating, so in this section, we need to use the standard deviation that applies to sample means, which is
. The fraction
is commonly called the standard error of the mean in order to distinguish clearly the standard deviation for a mean from the population standard deviation, σ.
2.
is normally distributed, that is,
~ N
.
3. When the population standard deviation
### Calculating the Confidence Interval
To construct a confidence interval estimate for an unknown population mean, we need data from a random sample. The steps to construct and interpret the confidence interval are as follows:
1. Calculate the sample mean,
from the sample data. Remember, in this section, we already know the population standard deviation, σ.
2. Find the z-score that corresponds to the confidence level.
3. Calculate the error bound EBM.
4. Construct the confidence interval.
5. If we denote the critical z-score by
, and the sample size by n, then the formula for the confidence interval with confidence level
, is given by
6. Write a sentence that interprets the estimate in the context of the situation in the problem. (Explain what the confidence interval means, in the words of the problem.)
We will first examine each step in more detail and then illustrate the process with some examples.
### Finding the z-Score for the Stated Confidence Level
When we know the population standard deviation, σ, we use a standard normal distribution to calculate the error bound EBM and construct the confidence interval. We need to find the value of z that puts an area equal to the confidence level (in decimal form) in the middle of the standard normal distribution Z ~ N(0, 1).
The confidence level, CL, is the area in the middle of the standard normal distribution. CL = 1 – α, so α is the area that is split equally between the two tails. Each of the tails contains an area equal to
.
The z-score that has an area to the right of
is denoted by
.
For example, when CL = 0.95, α = 0.05, and
= 0.025, we write
= 0.025.
The area to the right of z0.025 is 0.025 and the area to the left of z0.025 is 1 – 0.025 = 0.975.
, using a calculator, computer, or standard normal probability table.
Normal table (see appendices) shows that the probability for 0 to 1.96 is 0.47500, and so the probability to the right tail of the critical value 1.96 is 0.5 – 0.475 = 0.025
### Calculating the Margin of Error EBM
The error bound formula for an unknown population mean, μ, when the population standard deviation, σ, is known is
Margin of error =
### Constructing the Confidence Interval
The confidence interval estimate has the format
sample mean plus or minus the margin of error.
The graph gives a picture of the entire situation
CL +
+
= CL + α = 1.
### Writing the Interpretation
The interpretation should clearly state the confidence level (CL), explain which population parameter is being estimated (here, a population mean), and state the confidence interval (both endpoints): "We estimate with ___percent confidence that the true population mean (include the context of the problem) is between ___ and ___ (include appropriate units)."
Notice the difference in the confidence intervals calculated in and the following Try It exercise. These intervals are different for several reasons: they are calculated from different samples, the samples are different sizes, and the intervals are calculated for different levels of confidence. Even though the intervals are different, they do not yield conflicting information. The effects of these kinds of changes are the subject of the next section in this chapter.
### Changing the Confidence Level or Sample Size
### Working Backward to Find the Error Bound or Sample Mean
When we calculate a confidence interval, we find the sample mean, calculate the error bound, and use them to calculate the confidence interval. However, sometimes when we read statistical studies, the study may state the confidence interval only. If we know the confidence interval, we can work backward to find both the error bound and the sample mean.
2. From the upper value for the interval, subtract the sample mean,
3. Or, from the upper value for the interval, subtract the lower value. Then divide the difference by 2.
2. Subtract the error bound from the upper value of the confidence interval,
3. Or, average the upper and lower endpoints of the confidence interval.
Notice that there are two methods to perform each calculation. You can choose the method that is easier to use with the information you know.
### Calculating the Sample Size n
If researchers desire a specific margin of error, then they can use the error bound formula to calculate the required sample size. In this situation, we are given the desired margin of error, EBM, and we need to compute the sample size n.
The formula for sample size is n =
, found by solving the error bound formula for n. Always round up the value of n to the closest integer.
In this formula, z is the critical value
, corresponding to the desired confidence level. A researcher planning a study who wants a specified confidence level and error bound can use this formula to calculate the size of the sample needed for the study.
### References
Centers for Disease Control and Prevention. (n.d.). National health and nutrition examination survey. Retrieved from http://www.cdc.gov/nchs/nhanes.htm
Foothill De Anza Community College District. (n.d.). Headcount enrollment trends by student demographics ten-year fall trends to most recently completed fall. Retrieved from http://research.fhda.edu/factbook/FH_Demo_Trends/FoothillDemographicTrends.htm
Kuczmarski, R. J., et al. (2002). 2000 CDC growth charts for the United States: Methods and development. Retrieved from http://www.cdc.gov/growthcharts/2000growthchart-us.pdf
La, L., and; German, K. (n.d.). Cell phones with the highest radiation levels. CNET. Retrieved from http://reviews.cnet.com/cell-phone-radiation-levels/
U.S. Census Bureau. (n.d.). American FactFinder. Retrieved from http://factfinder2.census.gov/faces/nav/jsf/pages/searchresults.xhtml?refresh=t
U.S. Census Bureau. (2011). Mean income in the past 12 months (in 2011 inflation-adjusted dollars): 2011 American Community Survey 1-year estimates. Retrieved from http://factfinder2.census.gov/faces/tableservices/jsf/pages/productview.xhtml?pid=ACS_11_1YR_S1902&prodType=table
U.S. Federal Election Commission. (n.d.). Disclosure data catalog: Candidate summary report 2012. Retrieved from http://www.fec.gov/data/CandidateSummary.do?format=html&election_yr=2012
U.S. Federal Election Commission. (n.d.). Metadata description of candidate summary file. Retrieved from http://www.fec.gov/finance/disclosure/metadata/metadataforcandidatesummary.shtml
### Chapter Review
In this module, we learned how to calculate the confidence interval for a single population mean where the population standard deviation is known. When estimating a population mean, the margin of error is called the error bound for a population mean (EBM). A confidence interval has the general form
(lower bound, upper bound) = (point estimate – EBM, point estimate + EBM).
The calculation of EBM depends on the size of the sample and the level of confidence desired. The confidence level is the percentage of all possible samples that can be expected to include the true population parameter. As the confidence level increases, the corresponding EBM increases as well. As the sample size increases, the EBM decreases. By the central limit theorem,
Given a confidence interval, you can work backward to find the error bound (EBM) or the sample mean. To find the error bound, find the difference of the upper bound of the interval and the mean. If you do not know the sample mean, you can find the error bound by calculating half of the difference of the upper and lower bounds. To find the sample mean given a confidence interval, find the difference of the upper bound and the error bound. If the error bound is unknown, then average the upper and lower bounds of the confidence interval to find the sample mean.
Sometimes researchers know in advance that they want to estimate a population mean within a specific margin of error for a given level of confidence. In that case, solve the EBM formula for n to discover the size of the sample that is needed to achieve this goal:
### Formula Review
The distribution of sample means is normally distributed with mean equal to the population mean and standard deviation given by the population standard deviation divided by the square root of the sample size.
The general form for a confidence interval for a single population mean, known standard deviation, normal distribution is given by
(lower bound, upper bound) = (point estimate – EBM, point estimate + EBM)
EBM =
= the error bound for the mean, or the margin of error for a single population mean; this formula is used when the population standard deviation is known.
CL = confidence level, or the proportion of confidence intervals created that is expected to contain the true population parameter
α = 1 – CL = the proportion of confidence intervals that will not contain the population parameter
= the z-score with the property that the area to the right of the z-score is
; this is the z-score, used in the calculation of EBM, where α = 1 – CL.
n =
= the formula used to determine the sample size (n) needed to achieve a desired margin of error at a given level of confidence
General form of a confidence interval
(lower value, upper value) = (point estimate error bound, point estimate + error bound)
To find the error bound when you know the confidence interval,
error bound = upper value point estimate or error bound =
Single population mean, known standard deviation, normal distribution
Use the normal distribution for means; population standard deviation is known: EBM = z
The confidence interval has the format (
− EBM,
+ EBM).
Use the following information to answer the next five exercises: The standard deviation of the weights of elephants is known to be approximately 15 lb. We wish to construct a 95 percent confidence interval for the mean weight of newborn elephant calves. Fifty newborn elephants are weighed. The sample mean is 244 lb. The sample standard deviation is 11 lb.
Use the following information to answer the next seven exercises: The U.S. Census Bureau conducts a study to determine the time needed to complete the short form. The bureau surveys 200 people. The sample mean is 8.2 minutes. There is a known standard deviation of 2.2 minutes. The population distribution is assumed to be normal.
Use the following information to answer the next 10 exercises: A sample of 20 heads of lettuce was selected. Assume that the population distribution of head weight is normal. The weight of each head of lettuce was then recorded. The mean weight was 2.2 lb, with a standard deviation of 0.1 lb. The population standard deviation is known to be 0.2 lb.
Use the following information to answer the next 14 exercises: The mean age for all Foothill College students for a recent fall term was 33.2. The population standard deviation has been pretty consistent at 15. Suppose that 25 winter students were randomly selected. The mean age for the sample was 30.4. We are interested in the true mean age for winter Foothill College students. Let X = the age of a winter Foothill College student.
Construct a 95 percent confidence interval for the true mean age of winter Foothill College students by working out and then answering the next eight exercises.
### Homework
|
# Confidence Intervals
## A Single Population Mean Using the Student's t-Distribution
In practice, we rarely know the population standard deviation. In the past, when the sample size was large, this unknown number did not present a problem to statisticians. They used the sample standard deviation s as an estimate for σ and proceeded as before to calculate a confidence interval with close-enough results. However, statisticians ran into problems when the sample size was small. A small sample size caused inaccuracies in the confidence interval.
William S. Gosset (1876–1937) of the Guinness brewery in Dublin, Ireland, ran into this problem. His experiments with hops and barley produced very few samples. Just replacing σ with s did not produce accurate results when he tried to calculate a confidence interval. He realized that he could not use a normal distribution for the calculation; he found that the actual distribution depends on the sample size. This problem led him to discover what is called the Student's . The name comes from the fact that Gosset wrote under the pen name Student.
Up until the mid-1970s, some statisticians used the normal distribution approximation for large sample sizes and used the Student's t-distribution only for sample sizes of at most 30. With graphing calculators and computers, the practice now is to use the Student's t-distribution whenever s is used as an estimate for σ.
If you draw a simple random sample of size n from a population that has an approximately normal distribution with mean μ and unknown population standard deviation σ and calculate the t-score t =
, then the t-scores follow a Student's . The t-score has the same interpretation as the : It measures how far
is from its mean μ. For each sample size n, there is a different Student's t-distribution.
The degrees of freedom (, n - 1, are the sample size minus 1.
2. The graph for the Student's t-distribution is similar to the standard normal curve.
3. The mean for the Student's t-distribution is zero, and the distribution is symmetric about zero.
4. The Student's t-distribution has more probability in its tails than the standard normal distribution. shows the graphs of the student t-distribution for 1, 2 and 5 degrees of freedom: (v), compare to the standard normal distribution (in black).
5. The exact shape of the Student's t-distribution depends on the degrees of freedom. As the degrees of freedom increase, the graph of the Student's t-distribution becomes more like the graph of the standard normal distribution.
6. The underlying population of individual observations is assumed to be normally distributed with unknown population mean μ and unknown population standard deviation σ. The size of the underlying population is generally not relevant unless it is very small. If it is bell-shaped (normal), then the assumption is met and does not need discussion. Random sampling is assumed, but that is a completely separate assumption from normality.
Calculators and computers can easily calculate any Student's t-probabilities. The TI-83, 83+, and 84+ have a tcdf function to find the probability for given values of t. The grammar for the tcdf command is tcdf(lower bound, upper bound, degrees of freedom). However, for confidence intervals, we need to use inverse probability to find the value of t when we know the probability.
For the TI-84+, you can use the invT command on the DISTRibution menu. The invT command works similarly to the invnorm. The invT command requires two inputs: invT(area to the left, degrees of freedom). The output is the t-score that corresponds to the area we specified.
The TI-83 and 83+ do not have the invT command. (The TI-89 has an inverse T command.)
A probability table for the Student's t-distribution can also be used. The table gives critical t-values that correspond to the confidence level (column) and degrees of freedom (row). (The TI-86 does not have an invT program or command, so if you are using that calculator, you need to use a probability table for the Student's t-distribution.) When using a t-table, note that some tables are formatted to show the confidence level in the column headings, while the column headings in some tables may show only corresponding area in one or both tails.
A Student's t-table (see ) gives t-scores given the degrees of freedom and the right-tailed probability. The table is very limited. Calculators and computers can easily calculate any Student's t-probabilities.
If the population standard deviation is not known, the error bound for a population mean is
1.
,
2.
is the t-score with area to the right equal to
,
3. use df = n – 1 degrees of freedom, and
4. s = sample standard deviation.
The format for the confidence interval is
### References
Bloomberg Businessweek. (n.d.). Retrieved from http://www.businessweek.com/
Environmental Working Group. (n.d.). Human toxome project: Mapping the pollution in people. Retrieved from http://www.ewg.org/sites/humantoxome/participants/participant-group.php?group=in+utero%2Fnewborn
Federal Election Commission. (n.d.). Disclosure data catalog: 2012 leadership PACs and sponsors. Retrieved from http://www.fec.gov/data/index.jsp
Federal Election Commission. (n.d.). Metadata description of leadership PAC list. Retrieved from http://www.fec.gov/finance/disclosure/metadata/metadataLeadershipPacList.shtml
Forbes. (2013). Americas best small companies. Retrieved from http://www.forbes.com/best-small-companies/list/
Forbes. (n.d.). Retrieved from http://www.forbes.com/
Microsoft Bookshelf. (n.d.).
### Chapter Review
In many cases, the researcher does not know the population standard deviation, σ, of the measure being studied. In these cases, it is common to use the sample standard deviation, s, as an estimate of σ. The normal distribution creates accurate confidence intervals when σ is known, but it is not as accurate when s is used as an estimate. In this case, the Student’s t-distribution is much better. Define a t-score using the following formula:
The t-score follows the Student’s t-distribution with n – 1 degrees of freedom. The confidence interval under this distribution is calculated with EBM =
where
is the t-score with area to the right equal to
, s is the sample standard deviation, and n is the sample size. Use a table, calculator, or computer to find
for a given α.
### Formula Review
s = the standard deviation of sample values
is the formula for the t-score, which measures how far away a measure is from the population mean in the Student’s t-distribution.
df = n – 1; the degrees of freedom for a Student’s t-distribution, where n represents the size of the sample
T~t the random variable, T, has a Student’s t-distribution with df degrees of freedom
= the error bound for the population mean when the population standard deviation is unknown
is the t-score in the Student’s t-distribution with area to the right equal to
The general form for a confidence interval for a single mean, population standard deviation unknown, Student's t is given by (lower bound, upper bound)
= (point estimate – EBM, point estimate + EBM)
Use the following information to answer the next five exercises: A hospital is trying to cut down on emergency room wait times. It is interested in the amount of time patients must wait before being called back to be examined. An investigation committee randomly surveyed 70 patients. The sample mean was 1.5 hr, with a sample standard deviation of 0.5 hr.
Use the following information to answer the next six exercises: One hundred eight Americans were surveyed to determine the number of hours they spend watching television each month. It was revealed that they watch an average of 151 hours each month, with a standard deviation of 32 hours. Assume that the underlying population distribution is normal.
Use the following information to answer the next 13 exercises: The data in are the result of a random survey of 39 national flags (with replacement between picks) from various countries. We are interested in finding a confidence interval for the true mean number of colors on a national flag. Let X = the number of colors on a national flag.
Construct a 95 percent confidence interval for the true mean number of colors on national flags.
### Homework
Use the following information to answer the next two exercises: A quality control specialist for a restaurant chain takes a random sample of size 12 to check the amount of soda served in the 16-oz serving size. The sample mean is 13.30, with a sample standard deviation of 1.55. Assume the underlying population is normally distributed. |
# Confidence Intervals
## A Population Proportion
During an election year, we see articles in the newspaper that state confidence intervals in terms of proportions or percentages. For example, a poll for a particular candidate running for president might show that the candidate has 40 percent of the vote within 3 percentage points (if the sample is large enough). Often, election polls are calculated with 95 percent confidence, so the pollsters would be 95 percent confident that the true proportion of voters who favored the candidate would be between 0.37 and 0.43 (0.40 – 0.03, 0.40 + 0.03).
Investors in the stock market are interested in the true proportion of stocks that go up and down each week. Businesses that sell personal computers are interested in the proportion of households in the United States that own personal computers. Confidence intervals can be calculated for the true proportion of stocks that go up or down each week and for the true proportion of households in the United States that own personal computers.
The procedure to find the confidence interval, the sample size, the error bound for a population (, and the confidence level for a proportion is similar to that for the population mean, but the formulas are different.
How do you know you are dealing with a proportion problem? First, the data that you are collecting is categorical, consisting of two categories: Success or Failure, Yes or No. Examples of situations where you are the following trying to estimate the true population proportion are the following: What proportion of the population smoke? What proportion of the population will vote for candidate A? What proportion of the population has a college-level education?
The distribution of the sample proportions (based on samples of size n) is denoted by P′ (read “P prime”).
The central limit theorem for proportions asserts that the sample proportion distribution P′ follows a normal distribution with mean value p, and standard deviation
, where p is the population proportion and q = 1 - p.
The confidence interval has the form (p′ – EBP, p′ + EBP). EBP is error bound for the proportion.
p′ = the estimated proportion of successes (p′ is a point estimate for p, the true proportion.)
x = the number of successes
n = the size of the sample
The error bound for a proportion is
where q′ = 1 – p′.
This formula is similar to the error bound formula for a mean, except that the "appropriate standard deviation" is different. For a mean, when the population standard deviation is known, the appropriate standard deviation that we use is
. For a proportion, the appropriate standard deviation is
.
However, in the error bound formula, we use
as the standard deviation, instead of
.
In the error bound formula, the sample proportions . The estimated proportions p′ and q′ are used because p and q are not known. The sample proportions p′ and q′ are calculated from the data: p′ is the estimated proportion of successes, and q′ is the estimated proportion of failures.
The confidence interval can be used only if the number of successes np′ and the number of failures nq′ are both greater than five.
That is, in order to use the formula for confidence intervals for proportions, you need to verify that both
and
.
### Plus-Four Confidence Interval for p
There is a certain amount of error introduced into the process of calculating a confidence interval for a proportion. Because we do not know the true proportion for the population, we are forced to use point estimates to calculate the appropriate standard deviation of the sampling distribution. Studies have shown that the resulting estimation of the standard deviation can be flawed.
Fortunately, there is a simple adjustment that allows us to produce more accurate confidence intervals: We simply pretend that we have four additional observations. Two of these observations are successes, and two are failures. The new sample size, then, is n + 4, and the new count of successes is x + 2.
Computer studies have demonstrated the effectiveness of the plus-four confidence interval for . It should be used when the confidence level desired is at least 90 percent and the sample size is at least ten.
### Calculating the Sample Size n
If researchers desire a specific margin of error, then they can use the error bound formula to calculate the required sample size.
The margin of error formula for a population proportion is
1.
, where p′ is the sample proportion, q′ = 1 – p′, and n is the sample size.
2. Solving for n gives you an equation for the sample size.
3.
. This formula tells us that we can compute the sample size n required for a confidence level of
by taking the square of the critical value
, multiplying by the point estimate p′, and by q′ = 1 – p′ and finally dividing the result by the square of the margin of error. Always remember to round up the value of n.
### References
Jensen, T. (2013). Democrats, Republicans divided on opinion of music icons. Retrieved from http://www.publicpolicypolling.com/Day2MusicPoll.pdf
Madden, M., et al. (2013). Teens, social media, and privacy. Retrieved from http://www.pewinternet.org/Reports/2013/Teens-Social-Media-And-Privacy.aspx
Princeton Survey Research Associates International. (2012). 2012 teens and privacy management survey. Retrieved from http://www.pewinternet.org/~/media//Files/Questionnaire/2013/Methods%20and%20Questions_Teens%20and%20Social%20Media.pdf
Rasmussen Reports. (2013). 52% say big-time college athletics corrupt education process. Retrieved from http://www.rasmussenreports.com/public_content/lifestyle/sports/may_2013/52_say_big_time_college_athletics_corrupt_education_process
Saad, L. (2013). Three in four U.S. workers plan to work past retirement age. Retrieved from http://www.gallup.com/poll/162758/three-fourworkers-plan-work-past-retirement-age.aspx
The Field Poll. (n.d.). Retrieved from http://field.com/fieldpollonline/subscribers/
Zogby Analytics. (2013). New SUNYIT/Zogby analytics poll: Few Americans worry about emergency situations occurring in their community; Only one in three have an emergency plan; 70% support infrastructure investment for national security. Retrieved from http://www.zogbyanalytics.com/news/299-americans-neither-worried-norprepared-in-case-of-a-disaster-sunyit-zogby-analytics-poll
### Chapter Review
Some statistical measures, like many survey questions, measure qualitative rather than quantitative data. In this case, the population parameter being estimated is a proportion. It is possible to create a confidence interval for the true population proportion by following procedures similar to those used in creating confidence intervals for population means. The formulas are slightly different, but they follow the same reasoning.
Let p′ represent the sample proportion, x/n, where x represents the number of successes, and n represents the sample size. Let q′ = 1 – p′. Then the confidence interval for a population proportion is given by the following formula:
(lower bound, upper bound)
The plus–four method for calculating confidence intervals is an attempt to balance the error introduced by using estimates of the population proportion when calculating the standard deviation of the sampling distribution. Simply imagine four additional trials in the study; two are successes and two are failures. Calculate
, and proceed to find the confidence interval. When sample sizes are small, this method has been demonstrated to provide more accurate confidence intervals than the standard formula used for larger samples.
### Formula Review
p′ = x/n, where x represents the number of successes and n represents the sample size. The variable p′ is the sample proportion and serves as the point estimate for the true population proportion.
The variable p′ has a binomial distribution that can be approximated with the normal distribution shown here,
Confidence interval for a proportion:
provides the number of participants needed to estimate the population proportion with confidence 1 – α and margin of error EBP.
Use the normal distribution for a single population proportion
The confidence interval has the format (p′ – EBP, p′ + EBP).
is a point estimate for μ.
p′ is a point estimate for ρ.
s is a point estimate for σ.
Use the following information to answer the next two exercises: Marketing companies are interested in knowing the population percentage of women who make the majority of household purchasing decisions.
Use the following information to answer the next five exercises: Suppose a marketing company conducted a survey. It randomly surveyed 200 households and found that in 120 of them, the women made the majority of the purchasing decisions. We are interested in the population proportion of households where women make the majority of the purchasing decisions.
Use the following information to answer the next five exercises: Of 1,050 randomly selected adults, 360 identified themselves as manual laborers, 280 identified themselves as non-manual wage earners, 250 identified themselves as mid-level managers, and 160 identified themselves as executives. In the survey, 82 percent of manual laborers preferred trucks, 62 percent of non-manual wage earners preferred trucks, 54 percent of mid-level managers preferred trucks, and 26 percent of executives preferred trucks.
Use the following information to answer the next five exercises: A poll of 1,200 voters asked what the most significant issue was in the upcoming election. Sixty-five percent answered "the economy." We are interested in the population proportion of voters who believe the economy is the most important.
Use the following information to answer the next 16 exercises: The Ice Chalet offers dozens of different beginning ice-skating classes. All of the class names are put into a bucket. The 5 p.m., Monday night, ages 8 to 12, beginning ice-skating class is picked. In that class are 64 girls and 16 boys. Suppose that we are interested in the true proportion of girls, ages 8 to 12, in all beginning ice-skating classes at the Ice Chalet. Assume that the children in the selected class are a random sample of the population.
### Homework
Use the following information to answer the next three exercises: According to a Field Poll, 79 percent of California adults (actual results are 400 out of 506 surveyed) believe that education and our schools is one of the top issues facing California. We wish to construct a 90 percent confidence interval for the true proportion of California adults who believe that education and the schools is one of the top issues facing California.
Use the following information to answer the next two exercises: Five hundred eleven (511) homes in a certain southern California community are randomly surveyed to determine whether they meet minimal earthquake preparedness recommendations. One hundred seventy-three (173) of the homes surveyed meet the minimum recommendations for earthquake preparedness, and 338 do not. |
# Hypothesis Testing with One Sample
## Introduction
One job of a statistician is to make statistical inferences about populations based on samples taken from the population. Confidence intervals are one way to estimate a population parameter. Another way to make a statistical inference is to make a decision about a parameter. For instance, a car dealer advertises that its new small truck gets 35 miles per gallon, on average. A tutoring service claims that its method of tutoring helps 90 percent of its students get an A or a B. A company says that women managers in their company earn an average of $60,000 per year.
A statistician will make a decision about these claims. This process is called hypothesis testing. A hypothesis test involves collecting data from a sample and evaluating the data. Then, the statistician makes a decision as to whether or not there is sufficient evidence, based upon analyses of the data, to reject the null hypothesis.
In this chapter, you will conduct hypothesis tests on single means and single proportions.
You will also learn about the errors associated with these tests.
Hypothesis testing consists of two contradictory hypotheses or statements, a decision based on the data, and a conclusion. To perform a hypothesis test, a statistician will do the following:
|
# Hypothesis Testing with One Sample
## Null and Alternative Hypotheses
The actual test begins by considering two hypotheses. They are called the null hypothesis and the alternative hypothesis. These hypotheses contain opposing viewpoints.
H, the null hypothesis: a statement of no difference between sample means or proportions or no difference between a sample mean or proportion and a population mean or proportion. In other words, the difference equals 0.
H, the alternative hypothesis: a claim about the population that is contradictory to H and what we conclude when we reject H.
Since the null and alternative hypotheses are contradictory, you must examine evidence to decide if you have enough evidence to reject the null hypothesis or not. The evidence is in the form of sample data.
After you have determined which hypothesis the sample supports, you make a decision.
There are two options for a decision. They are reject H if the sample information favors the alternative hypothesis or do not reject H or decline to reject H if the sample information is insufficient to reject the null hypothesis.
Mathematical Symbols Used in H and H:
### Chapter Review
In a hypothesis test, sample data are evaluated in order to arrive at a decision about some type of claim. If certain conditions about the sample are satisfied, then the claim can be evaluated for a population. In a hypothesis test, we do the following:
### Formula Review
H and H are contradictory.
If α ≤ p-value, then do not reject H.
If α > p-value, then reject H.
α is preconceived. Its value is set before the hypothesis test starts. The p-value is calculated from the data.
### Homework
### References
Centers for Disease Control and Prevention. (n.d.). Physical activity facts. Retrieved from http://www.cdc.gov/healthyschools/physicalactivity/facts.htm
National Institute of Mental Health. (n.d.). Publications about depression. Retrieved from http://www.nimh.nih.gov/publicat/depression.cfm |
# Hypothesis Testing with One Sample
## Outcomes and the Type I and Type II Errors
When you perform a hypothesis test, there are four possible outcomes depending on the actual truth, or falseness, of the null hypothesis H and the decision to reject or not. The outcomes are summarized in the following table:
The four possible outcomes in the table are as follows:
Each of the errors occurs with a particular probability. The Greek letters α and β represent the probabilities.
α = probability of a Type I error = = probability of rejecting the null hypothesis when the null hypothesis is true.
β = probability of a Type II error = = probability of not rejecting the null hypothesis when the null hypothesis is false.
α and β should be as small as possible because they are probabilities of errors. They are rarely zero.
The Power of the Test is 1 – β. Ideally, we want a high power that is as close to one as possible. Increasing the sample size can increase the Power of the Test.
The following are examples of Type I and Type II errors.
### Chapter Review
In every hypothesis test, the outcomes are dependent on a correct interpretation of the data. Incorrect calculations or misunderstood summary statistics can yield errors that affect the results. A Type I error occurs when a true null hypothesis is rejected. A Type II error occurs when a false null hypothesis is not rejected.
The probabilities of these errors are denoted by the Greek letters α and β, for a Type I and a Type II error respectively. The power of the test, 1 – β, quantifies the likelihood that a test will yield the correct result of a true alternative hypothesis being accepted. A high power is desirable.
### Formula Review
α = probability of a Type I error = P(Type I error) = probability of rejecting the null hypothesis when the null hypothesis is true.
β = probability of a Type II error = P(Type II error) = probability of not rejecting the null hypothesis when the null hypothesis is false.
### Homework
|
# Hypothesis Testing with One Sample
## Distribution Needed for Hypothesis Testing
Earlier in the course, we discussed sampling distributions. Particular distributions are associated with hypothesis testing. Perform tests of a population mean using a normal distribution or a Student's . (Remember, use a Student's t-distribution when the population standard deviation is unknown and the distribution of the sample mean is approximately normal.) We perform tests of a population proportion using a normal distribution (usually n is large).
### Assumptions
When you perform a hypothesis test of a single population mean using a
Student's (often called a t-test), there are fundamental assumptions that need to be met in order for the test to work properly. Your data should be a simple random sample that comes from a population that is approximately normally distributed. You use the sample standard deviation to approximate the population standard deviation. Note that if the sample size is sufficiently large, a t-test will work even if the population is not approximately normally distributed.
When you perform a hypothesis test of a single population mean using a
normal distribution (often called a z-test), you take a simple random sample from the population. The population you are testing is normally distributed or your sample size is sufficiently large. You know the value of the population standard deviation which, in reality, is rarely known.
When you perform a hypothesis test of a single population proportion , you
take a simple random sample from the population. You must meet the conditions for a binomial distribution, which are the following: there are a certain number n of independent trials, the outcomes of any trial are success or failure, and each trial has the same probability of a success p. The shape of the binomial distribution needs to be similar to the shape of the normal distribution. To ensure this, the quantities np and nq must both be greater than five (np > 5 and nq > 5). Then the binomial distribution of a sample (estimated) proportion can be approximated by the normal distribution with μ = p and
.
Remember that q = 1 – p.
### Chapter Review
In order for a hypothesis test’s results to be generalized to a population, certain requirements must be satisfied.
When testing for a single population mean:
1. A Student's t-test should be used if the data come from a simple, random sample and the population is approximately normally distributed, or the sample size is large, with an unknown standard deviation.
2. The normal test will work if the data come from a simple, random sample and the population
is approximately normally distributed, or the sample size is large, with a known standard
deviation.
When testing a single population proportion use a normal test for a single population proportion if the data come from a simple, random sample, fill the requirements for a binomial distribution, and the mean number of success and the mean number of failures satisfy the conditions: np > 5 and nq > n where n is the sample size, p is the probability of a success, and q is the probability of a failure.
### Formula Review
If there is no given preconceived α, then use α = 0.05.
2. Single population mean, known population variance (or standard deviation): Normal
test.
3. Single population mean, unknown population variance (or standard deviation): Student's .
4. Single population proportion: Normal test.
5. For a single population mean, we may use a normal distribution with the following mean and standard deviation. Means:
and
6. For a single population proportion, we may use a normal distribution with the following mean and standard deviation. Proportions: µ = and
.
### Homework
|
# Hypothesis Testing with One Sample
## Rare Events, the Sample, and the Decision and Conclusion
Establishing the type of distribution, sample size, and known or unknown standard deviation can help you figure out how to go about a hypothesis test. However, there are several other factors you should consider when working out a hypothesis test.
### Rare Events
The thinking process in hypothesis testing can be summarized as follows: You want to test whether or not a particular property of the population is true. You make an assumption about the true population mean for numerical data or the true population proportion for categorical data. This assumption is the null hypothesis. Then you gather sample data that is representative of the population. From this sample data you compute the sample mean (or the sample proportion). If the value that you observe is very unlikely to occur (a rare event) if the null hypothesis is true, then you wonder why this is happening. A plausible explanation is that the null hypothesis is false.
For example, Didi and Ali are at a birthday party of a very wealthy friend. They hurry to be first in line to grab a prize from a tall basket that they cannot see inside because they will be blindfolded. There are 200 plastic bubbles in the basket, and Didi and Ali have been told that there is only one with a $100 bill. Didi is the first person to reach into the basket and pull out a bubble. Her bubble contains a $100 bill. The probability of this happening is = 0.005. Because this is so unlikely, Ali is hoping that what the two of them were told is wrong and there are more $100 bills in the basket. A rare event has occurred (Didi getting the $100 bill) so Ali doubts the assumption about only one $100 bill being in the basket.
### Using the Sample to Test the Null Hypothesis
After you collect data and obtain the test statistic (the sample mean, sample proportion, or other test statistic), you can determine the probability of obtaining that test statistic when the null hypothesis is true. This probability is called the .
When the p-value is very small, it means that the observed test statistic is very unlikely to happen if the null hypothesis is true. This gives significant evidence to suggest that the null hypothesis is false, and to reject it in favor of the alternative hypothesis. In practice, to reject the null hypothesis we want the p-value to be smaller than 0.05 (5 percent) or sometimes even smaller than 0.01 (1 percent).
### Decision and Conclusion
A systematic way to make a decision of whether to reject or not reject the null hypothesis
is to compare the p-value and a preset or preconceived α, also called the level of significance of the test. A preset α is the probability of a Type I error (rejecting the null hypothesis when the null hypothesis is true). It may or may not be given to you at the beginning of the problem.
When you make a decision to reject or not reject H, do as follows:
Conclusion: After you make your decision, write a thoughtful conclusion about the hypotheses in terms of the given problem.
### Chapter Review
When the probability of an event occurring is low, and it happens, it is called a rare event. Rare events are important to consider in hypothesis testing because they can inform your willingness not to reject or to reject a null hypothesis. To test a null hypothesis, find the p-value for the sample data and graph the results. When deciding whether or not to reject the null the hypothesis, keep these two parameters in mind:
Use the following information to answer the next seven exercises: Suppose that a recent article stated that the mean time students spend doing homework each week is 2.5 hours. A study was then done to see if the mean time has increased in the new century. A random sample of 26 students was taken. The mean length of time they did homework each week was three hours with a standard deviation of 1.8 hours. Suppose that it is somehow known that the population standard deviation is 1.5. Conduct a hypothesis test to determine if the mean length of time doing homework each week has increased. Assume the distribution of homework times is approximately normal.
### Homework
|
# Hypothesis Testing with One Sample
## Additional Information and Full Hypothesis Test Examples
1. In a hypothesis test problem, you may see words such as "the level of significance is 1 percent". The "1 percent" is the preconceived or preset α.
2. The statistician setting up the hypothesis test selects the value of α to use before collecting the sample data.
3. If no level of significance is given, a common standard to use is
4. When you calculate the p-value and draw the picture, the p-value is the area in the left tail, the right tail, or split evenly between the two tails. For this reason, we call the hypothesis test left, right, or two tailed.
5. The alternative hypothesis, , tells you if the test is left, right, or two-tailed. It is the key to conducting the appropriate test.
6. H never has a symbol that contains an equal sign.
7. Thinking about the meaning of the : A data analyst should have more confidence that he made the correct decision to reject the null hypothesis with a smaller p-value (for example, 0.001 as opposed to 0.04) even if using the 0.05 level for alpha. Similarly, for a large p-value such as 0.4, as opposed to a p-value of 0.056 (alpha = 0.05 is less than either number), a data analyst should have more confidence that she made the correct decision in not rejecting the null hypothesis. This makes the data analyst use judgment rather than mindlessly applying rules.
The following examples illustrate a left-, right-, and two-tailed test.
### Full Hypothesis Test Examples
The next example is a poem written by a statistics student named Nicole Hart. The solution to the problem follows the poem. Notice that the hypothesis test is for a single population proportion. This means that the null and alternate hypotheses use the parameter p. The distribution for the test is normal. The estimated proportion p′ is the proportion of fleas killed to the total fleas found on Fido. This is sample information. The problem gives a preconceived α = 0.01, for comparison, and a 95 percent confidence interval computation. The poem is clever and humorous, so please enjoy it!
### Chapter Review
The hypothesis test itself has an established process. This can be summarized as follows:
Notice that in performing the hypothesis test, you use α and not β. β is needed to help determine the sample size of the data that are used in calculating the p-value. Remember that the quantity 1 – β is called the Power of the Test. A high power is desirable. If the power is too low, statisticians typically increase the sample size while keeping α the same. If the power is low, the null hypothesis might not be rejected when it should be.
### Homework
For each of the word problems, use a solution sheet to do the hypothesis test. The solution sheet is found in
Hypothesis testing: For the following 10 exercises, answer each question.
1. State the null and alternate hypotheses.
2. State the p-value.
3. State alpha.
4. What is your decision?
5. Write a conclusion.
6. Answer any other questions asked in the problem.
### References
American Automobile Association. (n.d.). Retrieved from www.aaa.com
American Library Association. (n.d.). Retrieved from www.ala.org
Amit Schitai. (n.d.). Data.
Bureau of Labor Statistics. (n.d.). Occupational employment statistics. Retrieved from http://www.bls.gov/oes/current/oes291111.htm
Centers for Disease Control and Prevention. (n.d.). Retrieved from www.cdc.gov
De Anza College. (2006). Foothill-De Anza Community College District. Retrieved from http://research.fhda.edu/factbook/DAdemofs/Fact_sheet_da_2006w.pdf
Federal Bureau of Investigation. (n.d.). Uniform crime reports and index of crime in Daviess in the state of Kentucky enforced by Daviess County from 1985 to 2005. Retrieved from http://www.disastercenter.com/kentucky/crime/3868.htm
Gallup. (n.d.). Retrieved from www.gallup.com
Johansen, C., Boice, Jr. J., McLaughlin, J., & Olsen, J. (2001, Feb. 7). Cellular telephones and cancera nationwide cohort study in Denmark. Journal of National Cancer Institute, 93(3), 2037. Retrieved from http://www.ncbi.nlm.nih.gov/pubmed/11158188
La Leche League International. (n.d.). Retrieved from http://www.lalecheleague.org/Law/BAFeb01.html
Online Learning Consortium. (2005 Nov.). Growing by degrees: Online education in the United States, 2005. Newburyport, MA: Allen, I. E., & Seaman, J. Available at http://files.eric.ed.gov/fulltext/ED530062.pdf
Toastmasters International. (n.d.). Retrieved from http://toastmasters.org/artisan/detail.asp?CategoryID=1&SubCategoryID=10&ArticleID=429&Page=1
U.S. Census Bureau. (n.d.). Language use. Retrieved from https://www.census.gov/topics/population/language-use.html
U.S. Census Bureau. (n.d.). QuickFacts. Retrieved from https://www.census.gov/quickfacts/table/PST045216/00
U.S. Department of Energy. (n.d.). Retrieved from http://energy.gov
Weather Underground. (n.d.). Retrieved from www.wunderground.com |
# Hypothesis Testing with Two Samples
## Introduction
Studies often compare two groups. For example, researchers are interested in the effect aspirin has in preventing heart attacks. Over the last few years, newspapers and magazines have reported various aspirin studies involving two groups. Typically, one group is given aspirin and the other group is given a placebo. Then, the heart attack rate is studied over several years.
There are other situations that deal with the comparison of two groups. For example, studies compare various diet and exercise programs. Politicians compare the proportion of individuals from different income brackets who might vote for them. Students are interested in whether the SAT or GRE preparatory courses really help raise their scores.
You have learned to conduct hypothesis tests on single means and single proportions. You will expand upon that in this chapter. You will compare two means or two proportions to each other. The general procedure is the same, just expanded.
To compare two means or two proportions, you work with two groups. The groups are classified as independent groups or matched pairs. Independent groups consist of two samples that are independent, that is, sample values selected from one population are not related in any way to sample values selected from the other population. Matched pairs consist of two samples that are dependent. The parameter tested using matched pairs is the population mean. The parameters tested using independent groups are either population means or population proportions.
This chapter deals with the following hypothesis tests:
1. Independent groups (samples are independent)
2. Matched or paired samples (samples are dependent)
|
# Hypothesis Testing with Two Samples
## Two Population Means with Unknown Standard Deviations
1. The two independent samples are simple random samples from two distinct populations.
2. For the two distinct populations
The comparison of two population means is very common. A difference between the two samples depends on both the means and the standard deviations. Very different means can occur by chance if there is great variation among the individual samples. To account for the variation, we take the difference of the sample means,
, and divide by the standard error to standardize the difference. The result is a t-score test statistic.
Because we do not know the population standard deviations, we estimate them using the two sample standard deviations from our independent samples. For the hypothesis test, we calculate the estimated standard deviation, or standard error, of the difference in sample means,
The standard error is calculated as follows:
The test statistic (t-score) is calculated as follows:
The number of degrees of freedom ( requires a somewhat complicated calculation. However, a computer or calculator calculates it easily. The df are not always a whole number. The test statistic calculated previously is approximated by the Student’s t-distribution with df as follows:
Degrees of freedom
When both sample sizes n1 and n2 are five or larger, the Student’s t approximation is very good. Notice that the sample variances (s1)2 and (s2)2 are not pooled. (If the question comes up, do not pool the variances.)
Cohen’s Standards for Small, Medium, and Large Effect SizesCohen’s is a measure of effect size based on the differences between two means. Cohen’s d, named for U.S. statistician Jacob Cohen, measures the relative strength of the differences between the means of two populations based on sample data. The calculated value of effect size is then compared to Cohen’s standards of small, medium, and large effect sizes.
Cohen’s d is the measure of the difference between two means divided by the pooled standard deviation:
where
.
### References
Baseball-Almanac. (2013). World series history. Retrieved from http://www.baseball-almanac.com/ws/wsmenu.shtml
Graduating Engineer + Computer Careers. (n.d.). Retrieved from http://www.graduatingengineer.com
Microsoft Bookshelf. (n.d.).
Nasdaq. (n.d.). Sectoring by industry groups. Retrieved from http://www.nasdaq.com/markets/barchart-sectors.aspx
U.S. Senate. (n.d.). Retrieved from www.senate.gov
Wikipedia. (n.d.). List of current United States Senators by age. Retrieved from http://en.wikipedia.org/wiki/List_of_current_United_States_Senators_by_age
### Chapter Review
Two population means from independent samples where the population standard deviations are not known
1. Random variable:
= the difference of the sampling means
2. Distribution: Student’s t-distribution with degrees of freedom (variances not pooled)
### Formula Review
Standard error: SE =
Test statistic (t-score): t =
Degrees of freedom:
where:
s1 and s2 are the sample standard deviations, and n1 and n2 are the sample sizes.
and
are the sample means.
Cohen’s d is the measure of effect size:
where
Use the following information to answer the next 15 exercises. Indicate if the hypothesis test is for
1. independent group means, population standard deviations, and/or variances known,
2. independent group means, population standard deviations, and/or variances unknown,
3. matched or paired samples,
4. single mean,
5. two proportions, or
6. single proportion.
Use the following information to answer the next three exercises: A study is done to determine which of two soft drinks has more sugar. There are 13 cans of Beverage A in a sample and six cans of Beverage B. The mean amount of sugar in Beverage A is 36 grams with a standard deviation of 0.6 grams. The mean amount of sugar in Beverage B is 38 grams with a standard deviation of 0.8 grams. The researchers believe that Beverage B has more sugar than Beverage A, on average. Both populations have normal distributions.
Use the following information to answer the next 12 exercises. The U.S. Centers for Disease Control reports that the mean life expectancy was 47.6 years for whites born in 1900 and 33.0 years for nonwhites. Suppose that you randomly survey death records for people born in 1900 in a certain county. Of the 124 whites, the mean life span was 45.3 years with a standard deviation of 12.7 years. Of the 82 nonwhites, the mean life span was 34.1 years with a standard deviation of 15.6 years. Conduct a hypothesis test to see if the mean life spans in the county were the same for whites and nonwhites.
### Homework
DIRECTIONS: For each of the word problems, use a solution sheet to do the hypothesis test. The solution sheet is found in
Use the information from
Use the following information to answer the next two exercises. The Eastern and Western Major League Soccer conferences have a new Reserve Division that allows new players to develop their skills. Data for a randomly picked date showed the following annual goals.
Conduct a hypothesis test to answer the next two exercises. |
# Hypothesis Testing with Two Samples
## Two Population Means with Known Standard Deviations
Even though this situation is not likely (knowing the population standard deviations), the following example illustrates hypothesis testing for independent means, known population standard deviations. The sampling distribution for the difference between the means is normal, and both populations must be normal. The random variable is
. The normal distribution has the following format:
Normal distribution is
### References
Centers for Disease Control and Prevention. (2008, July 18). State-specific prevalence of obesity among adultsUnited States, 2007. MMWR, 57(28), 765768. Retrieved from http://www.cdc.gov/mmwr/preview/mmwrhtml/mm5728a1.htm
Federal Bureau of Investigation. (n.d.). Texas Crime Rates 19601012. Available at http://www.disastercenter.com/crime/txcrime.htm
Hinduja, S. (2013). Sexting Research and Gender Differences. Cyberbulling Research Center. Retrieved from http://cyberbullying.us/blog/sexting-research-and-gender-differences/
Humes, K. R., Jones, N. A., & Ramirez, R. R. (2011 March). Overview of race and Hispanic origin: 2010 (2010 Census Briefs). Washington, DC: U.S. Census Bureau. Available online at http://www.census.gov/prod/cen2010/briefs/c2010br-02.pdf
Smith, A. (2011, July 11). 35% of American adults own a smartphone. Pew Internet. Available online at http://www.pewinternet.org/~/media/Files/Reports/2011/PIP_Smartphones.pdf
Visually. (2013). Smart phone users, by the numbers. Retrieved from http://visual.ly/smart-phone-users-numbers
### Chapter Review
A hypothesis test of two population means from independent samples where the population standard deviations are known (typically approximated with the sample standard deviations) will have these characteristics:
### Formula Review
Normal distribution:
.
Generally,
Test statistic (z-score):
Generally,
where
σ1 and σ2 are the known population standard deviations, n1 and n2 are the sample sizes,
and
are the sample means, and μ1 and μ2 are the population means.
Use the following information to answer the next five exercises. The mean speeds of fastball pitches from two different baseball pitchers are to be compared. A sample of 14 fastball pitches is measured from each pitcher. The populations have normal distributions. shows the result. Scouters believe that Rodriguez pitches a speedier fastball.
Use the following information to answer the next five exercises. A researcher is testing the effects of plant food on plant growth. Nine plants have been given the plant food. Another nine plants have not been given the plant food. The heights of the plants are recorded after eight weeks. The populations have normal distributions. The following table is the result. The researcher thinks the food makes the plants grow taller.
Use the following information to answer the next five exercises. Two metal alloys are being considered as material for ball bearings. The mean melting point of the two alloys is to be compared. Fifteen pieces of each metal are being tested. Both populations have normal distributions. The following table is the result. It is believed that Alloy Zeta has a different melting point.
### Homework
DIRECTIONS: For each of the word problems, use a solution sheet to do the hypothesis test. The solution sheet is found in |
# Hypothesis Testing with Two Samples
## Comparing Two Independent Population Proportions
When conducting a hypothesis test that compares two independent population proportions, the following characteristics should be present:
1. The two independent samples are simple random samples that are independent.
2. The number of successes is at least five, and the number of failures is at least five, for each of the samples.
3. Growing literature states that the population must be at least 10 or 20 times the size of the sample. This keeps each population from being over-sampled and causing incorrect results.
Comparing two proportions, like comparing two means, is common. If two estimated proportions are different, it may be due to a difference in the populations or it may be due to chance. A hypothesis test can help determine if a difference in the estimated proportions reflects a difference in the population proportions.
The difference of two proportions follows an approximate normal distribution. Generally, the null hypothesis states that the two proportions are the same. That is, H: p = p. To conduct the test, we use a pooled proportion, p.
The pooled proportion is calculated as follows:
The distribution for the differences is
The test statistic (z-score) is
### References
American Cancer Society. (n.d.). Retrieved from http://www.cancer.org/index
Centers for Disease Control and Prevention. (n.d.). West Nile virus. Retrieved from http://www.cdc.gov/ncidod/dvbid/westnile/index.htm
Chancellors Office, California Community Colleges. (1994, Nov.).
Educational Resources. (n.d.).
Gallup. (2013). State of the states. Retrieved from http://www.gallup.com/poll/125066/State-States.aspx?ref=interactive
Hilton Hotels. (n.d.). Retrieved from http://www.hilton.com
Hyatt Hotels. Retrieved from http://hyatt.com
San Jose Museum of Art. (n.d.). Whitney exhibit (on loan).
U.S. Department of Health and Human Services. (n.d). Statistics. Retrieved from https://www.hhs.gov/
### Chapter Review
Test of two population proportions from independent samples
### Formula Review
Pooled proportion: p =
Distribution for the differences:
where the null hypothesis is H: p = p or H: p – p = 0
Test statistic (z-score):
where the null hypothesis is H: p = p or H: p − p = 0
and where
p′
and p′
are the sample proportions, p and p are the population proportions,
P is the pooled proportion, and n and n are the sample sizes.
Use the following information for the next five exercises. Two types of phone operating system are being tested to determine if there is a difference in the proportions of system failures (crashes). Fifteen out of a random sample of 150 phones with OS1 had system failures within the first eight hours of operation. Nine out of another random sample of 150 phones with OS2 had system failures within the first eight hours of operation. OS2 is believed to be more stable (have fewer crashes) than OS1.
Use the following information to answer the next 12 exercises. In the recent U.S. Census, 3 percent of the U.S. population reported being of two or more races. However, the percent varies tremendously from state to state. Suppose that two random surveys are conducted. In the first random survey, out of 1,000 North Dakotans, only 9 people reported being of two or more races. In the second random survey, out of 500 Nevadans, 17 people reported being of two or more races. Conduct a hypothesis test to determine if the population percents are the same for the two states or if the percent for Nevada is statistically higher than for North Dakota.
### Homework
DIRECTIONS: For each of the word problems, use a solution sheet to do the hypothesis test. The solution sheet is found in
Use the following information to answer the next three exercises. Neuroinvasive West Nile virus is a severe disease that affects a person’s nervous system. It is spread by the Culex species of mosquito. In the United States in 2010, there were 629 reported cases of neuroinvasive West Nile virus out of a total of 1,021 reported cases, and there were 486 neuroinvasive reported cases out of a total of 712 cases reported in 2011. Is the 2011 proportion of neuroinvasive West Nile virus cases more than the 2010 proportion of neuroinvasive West Nile virus cases? Using a 1 percent level of significance, conduct an appropriate hypothesis test.
|
# Hypothesis Testing with Two Samples
## Matched or Paired Samples (Optional)
When using a hypothesis test for matched or paired samples, the following characteristics should be present:
1. Simple random sampling is used.
2. Sample sizes are often small.
3. Two measurements (samples) are drawn from the same pair of individuals or objects.
4. Differences are calculated from the matched or paired samples.
5. The differences form the sample that is used for the hypothesis test.
6. Either the matched pairs have differences that come from a population that is normal or the number of differences is sufficiently large so that distribution of the sample mean of differences is approximately normal.
In a hypothesis test for matched or paired samples, subjects are matched in pairs and differences are calculated. The differences are the data. The population mean for the differences, μ, is then tested using a Student’s-t test for a single population mean with n – 1 degrees of freedom, where n is the number of differences.
The test statistic (t-score) is
### Chapter Review
A hypothesis test for matched or paired samples (t-test) has these characteristics:
1. Test the differences by subtracting one measurement from the other measurement
2. Random variable:
= mean of the differences.
3. Distribution: Student’s t distribution with n – 1 degrees of freedom.
4. If the number of differences is small (less than 30), the differences must follow a normal distribution.
5. Two samples are drawn from the same set of objects.
6. Samples are dependent.
### Formula Review
Test statistic (t-score): t =
where:
is the mean of the sample differences, μd is the mean of the population differences, s is the sample standard deviation of the differences, and n is the sample size.
Use the following information to answer the next five exercises. A study was conducted to test the effectiveness of a software patch in reducing system failures over a six-month period. Results for randomly selected installations are shown in . The before value is matched to an after value, and the differences are calculated. The differences have a normal distribution. Test at the 1 percent significance level.
Use the following information to answer next five exercises. A study was conducted to test the effectiveness of a juggling class. Before the class started, six subjects juggled as many balls as they could at once. After the class, the same six subjects juggled as many balls as they could. The differences in the number of balls are calculated. The differences have a normal distribution. Test at the 1 percent significance level.
Use the following information to answer the next five exercises. A doctor wants to know if a blood pressure medication is effective. Six subjects have their blood pressures recorded. After twelve weeks on the medication, the same six subjects have their blood pressure recorded again. For this test, only systolic pressure is of concern. Test at the 1 percent significance level.
### Homework
DIRECTIONS: For each of the word problems, use a solution sheet to do the hypothesis test. The solution sheet is found in
Use the following information to answer the next two exercises. A new preventative medication was tried on a group of 224 patients who had the same risk factors for a disease. 45 patients developed the disease after four years. In a control group of 224 patients, 68 developed the disease after four years. We want to test whether the method of treatment reduces the proportion of patients who develop the disease after four years.
Let the subscript t = treated patient and ut = untreated patient.
Use the following information to answer the next two exercises. An experiment is conducted to show that blood pressure can be consciously reduced in people trained in a biofeedback exercise program. Six subjects were randomly selected, and blood pressure measurements were recorded before and after the training. The difference between blood pressures was calculated (after – before), producing the following results:
= −10.2 s = 8.4. Using the data, test the hypothesis that the blood pressure has decreased after the training.
### Bringing It Together
Use the following information to answer the next 10 exercises. Indicate which of the following choices best identifies the hypothesis test.
1. Independent group means, population standard deviations and/or variances known
2. Independent group means, population standard deviations and/or variances unknown
3. Matched or paired samples
4. Single mean
5. Two proportions
6. Single proportion |
# The Chi-Square Distribution
## Introduction
Have you ever wondered if lottery numbers were evenly distributed or if some numbers occurred with a greater frequency? How about if the types of movies people preferred were different across different age groups? What about if a coffee machine was dispensing approximately the same amount of coffee each time? You could answer these questions by conducting a hypothesis test.
You will now study a new distribution, one that is used to determine the answers to such questions. This distribution is called the chi-square distribution.
In this chapter, you will learn the three major applications of the chi-square distribution:
|
# The Chi-Square Distribution
## Facts About the Chi-Square Distribution
The notation for the chi-square distribution is
where df = degrees of freedom, which depends on how chi-square is being used. If you want to practice calculating chi-square probabilities then use df = n – 1. The degrees of freedom for the three major uses are calculated differently.
For the χ distribution, the population mean is μ = df, and the population standard deviation is
.
The random variable is shown as χ, but it may be any uppercase letter.
The random variable for a chi-square distribution with k degrees of freedom is the sum of k independent, squared standard normal variables is
χ2 = (Z1)2 + (Z2)2 + ... + (Z)2, where the following are true:
1. The curve is nonsymmetrical and skewed to the right.
2. There is a different chi-square curve for each df.
3. The test statistic for any test is always greater than or equal to zero.
4. When df > 90, the chi-square curve approximates the normal distribution. For X ~
, the mean, μ = df = 1,000 and the standard deviation, σ =
= 44.7. Therefore, X ~ N(1,000, 44.7), approximately.
5. The mean, μ, is located just to the right of the peak.
### References
Parade Magazine. (n.d.). Retrieved from https://parade.com/
Santa Clara County Public Health Department. (2011, May). HIV/AIDS epidemiology Santa Clara County. Retrieved from http://sccgov.iqm2.com/Citizens/FileOpen.aspx?Type=4&ID=32762
### Chapter Review
The chi-square distribution is a useful tool for assessment in a series of problem categories. These problem categories include primarily (i) whether a data set fits a particular distribution, (ii) whether the distributions of two populations are the same, (iii) whether two events might be independent, and (iv) whether there is a different variability than expected within a population.
An important parameter in a chi-square distribution is the degrees of freedom df in a given problem. The random variable in the chi-square distribution is the sum of squares of df standard normal variables, which must be independent. The key characteristics of the chi-square distribution also depend directly on the degrees of freedom.
The chi-square distribution curve is skewed to the right, and its shape depends on the degrees of freedom df. For df > 90, the curve approximates the normal distribution. Test statistics based on the chi-square distribution are always greater than or equal to zero. Such application tests are almost always right-tailed tests.
### Formula Review
χ2 = (Z1)2 + (Z2)2 + . . . (Z)2
chi-square distribution random variable
μ = df chi-square distribution population mean
chi-square distribution population standard deviation
### Homework
Decide whether the following statements are true or false. |
# The Chi-Square Distribution
## Goodness-of-Fit Test
In this type of hypothesis test, you determine whether the data fit a particular distribution. For example, you may suspect your unknown data fit a binomial distribution. You use a chi-square test, meaning the distribution for the hypothesis test is chi-square, to determine if there is a fit. The null and the alternative hypotheses for this test may be written in sentences or may be stated as equations or inequalities.
The test statistic for a goodness-of-fit test is:
where
1. O = observed values (data),
2. E = expected values (from theory), and
3. k = the number of different data cells or categories.
The observed values are the data values, and the expected values are the values you would expect to get if the null hypothesis were true. There are n terms of the form
.
The number of degrees of freedom is df = (number of categories – 1).
The goodness-of-fit test is almost always right-tailed. If the observed values and the corresponding expected values are not close to each other, then the test statistic can get very large and will be way out in the right tail of the chi-square curve.
### References
College Board. (n.d.). Retrieved from http://www.collegeboard.com
Ma, Y., et al. (2003). Association between eating patterns and obesity in a free-living US adult population. American Journal of Epidemiology 158(1), 8592.
Ogden, C. L., et al. (2012, January). Prevalence of obesity in the United States, 20092010 (NCHS Data Brief No. 82). Hyattsville, MD: National Center for Health Statistics. Retrieved from http://www.cdc.gov/nchs/data/databriefs/db82.pdf
Stevens, B. J. (n.d.). Multi-family and commercial solid waste and recycling survey. Arlington County, VA. Retrieved from http://www.arlingtonva.us/departments/EnvironmentalServices/SW/file84429.pdf
U.S. Census Bureau. (n.d.). Current population reports. Retrieved from https://www.census.gov/main/www/cprs.html
U.S. Census Bureau. (n.d). Retrieved from https://www.census.gov/
### Chapter Review
To assess whether a data set fits a specific distribution, you can apply the goodness-of-fit hypothesis test that uses the chi-square distribution. The null hypothesis for this test states that the data come from the assumed distribution. The test compares observed values against the values you would expect to have if your data followed the assumed distribution. The test is almost always right-tailed. Each observation or cell category must have an expected value of at least five.
### Formula Review
goodness-of-fit test statistic where
O: observed values
E: expected values
k: number of different data cells or categories
df = k − 1 degrees of freedom
Determine the appropriate test to be used in the next three exercises.
Use the following information to answer the next five exercises. A teacher predicts the distribution of grades on the final exam. The predictions are shown in .
The actual distribution for a class of 20 is in .
Use the following information to answer the next nine exercises. The cumulative number of cases of a chronic disease reported for Santa Clara County is broken down by ethnicity as in .
The percentage of each ethnic group in Santa Clara County is as in .
### Homework
For each problem, use a solution sheet to solve the hypothesis test problem. Go to
Use the following information to answer the next two exercises. The columns in contain the Race/Ethnicity of U.S. Public Schools for a recent year, the percentages for the Advanced Placement Examinee Population for that class, and the Overall Student Population. Suppose the right column contains the results of a survey of 1,000 local students from that year who took an AP exam.
Use the following information to answer the next two exercises. UCLA conducted a survey of more than 263,000 college freshmen from 385 colleges in fall 2005. The results of students’ expected majors by gender were reported in . Suppose a survey of 5,000 graduating females and 5,000 graduating males was done as a follow-up last year to determine what their actual majors were. The results are shown in the tables for and . The second column in each table does not add to 100 percent because of rounding.
Read the statement and decide whether it is true or false. |
# The Chi-Square Distribution
## Test of Independence
Tests of independence involve using a contingency table of observed (data) values.
The test statistic for a test of independence is similar to that of a goodness-of-fit test
where
1. O = observed values,
2. E = expected values,
3. i = the number of rows in the table, and
4. j = the number of columns in the table.
There are terms of the form
.
A test of independence determines whether two factors are independent. You first encountered the term independence in Probability Topics. As a review, consider the following example.
### References
Harris Interactive. (n.d.). Retrieved from http://www.statisticbrain.com/favorite-flavor-of-ice-cream/
Statistics Brain. (2016, June 29). Youngest online entrepreneurs list. Retrieved from http://www.statisticbrain.com/youngest-online-entrepreneur-list
### Chapter Review
To assess whether two factors are independent, you can apply the test of independence that uses the chi-square distribution. The null hypothesis for this test states that the two factors are independent. The test compares observed values to expected values. The test is right-tailed. Each observation or cell category must have an expected value of at least five.
### Formula Review
2. The number of degrees of freedom is equal to (number of columns–1)(number of rows–1).
3. The test statistic is
where
O = observed values,
E = expected values,
i = the number of rows in the
table, and
j = the number of columns in
the table.
4. If the null hypothesis is true, the expected number
.
Determine the appropriate test to be used in the next three exercises.
Use the following information to answer the next seven exercises: Transit Railroads is interested in the relationship between travel distance and the ticket class purchased. A random sample of 200 passengers is taken. shows the results. The railroad wants to know if a passenger’s choice in ticket class is independent of the distance the passenger must travel.
Use the following information to answer the next ten exercises. An article in the New England Journal of Medicine discussed a study on people who used a certain product in California and Hawaii. In one part of the report, the self-reported ethnicity and product-use levels per day were given. Of the people using the product at most 10 times per day, there were 9,886 African Americans, 2,745 Native Hawaiians, 12,831 Latinos, 8,378 Japanese Americans, and 7,650 whites. Of the people using the product 11 to 20 times per day, there were 6,514 African Americans, 3,062 Native Hawaiians, 4,932 Latinos, 10,680 Japanese Americans, and 9,877 whites. Of the people using the product 21 to 30 times per day, there were 1,671 African Americans, 1,419 Native Hawaiians, 1,406 Latinos, 4,715 Japanese Americans, and 6,062 whites. Of the people using the product at least 31 times per day, there were 759 African Americans, 788 Native Hawaiians, 800 Latinos, 2,305 Japanese Americans, and 3,970 whites.
State the decision and conclusion (in a complete sentence) for the following levels of α.
### Homework
For each problem, use a solution sheet to solve the hypothesis test problem. Go to
Read the statement and decide whether it is true or false. |
# The Chi-Square Distribution
## Test for Homogeneity
The goodness-of-fit test can be used to decide whether a population fits a given distribution, but it will not suffice to decide whether two populations follow the same unknown distribution. A different test, called the test for homogeneity, can be used to draw a conclusion about whether two populations have the same distribution. To calculate the test statistic for a test for homogeneity, follow the same procedure as with the test of independence.
HypothesesH: The distributions of the two populations are the same.
H: The distributions of the two populations are not the same.
Test StatisticUse a test statistic. It is computed in the same way as the test for independence.
Degrees of freedom (df = number of columns – 1
RequirementsAll values in the table must be greater than or equal to five.
Common UsesComparing two populations. For example: men vs. women, before vs. after, east vs. west. The variable is categorical with more than two possible response values.
### References
Bielick, S. (2008, December). 1.5 million homeschooled students in the United States in 2007 (NCES 2009030).
Washington, DC: National Center for Education Statistics. Retrieved from http://nces.ed.gov/pubsearch/pubsinfo.asp?pubid=2009030
Bielick, S. (2008, December). 1.5 million homeschooled students in the United States in 2007supplemental tables (NCES 2009030). Washington, DC: National Center for Education Statistics. Retrieved from http://nces.ed.gov/pubs2009/2009030_sup.pdf
Insurance Institute for Highway Safety. (n.d.). Ratings. Retrieved from www.iihs.org/iihs/ratings
World Bank Group. (2014). Energy use (kg of oil equivalent per capita). Retrieved from http://data.worldbank.org/indicator/EG.USE.PCAP.KG.OE/countries
### Chapter Review
To assess whether two data sets are derived from the same distribution, which need not be known, you can apply the test for homogeneity that uses the chi-square distribution. The null hypothesis for this test states that the populations of the two data sets come from the same distribution. The test compares the observed values against the expected values if the two populations followed the same distribution. The test is right-tailed. Each observation or cell category must have an expected value of at least five.
### Formula Review
Homogeneity test statistic where O = observed values E = expected values i = number of rows in data contingency table j = number of columns in data contingency table
df = (i −1)(j −1) degrees of freedom
Use the following information to answer the next five exercises. Do private practice doctors and hospital doctors have the same distribution of working hours? Suppose that a sample of 100 private practice doctors and 150 hospital doctors are selected at random and asked about the number of hours a week they work. The results are shown in .
### Homework
For each word problem, use a solution sheet to solve the hypothesis test problem. Go to |
# The Chi-Square Distribution
## Comparison of the Chi-Square Tests
You have seen the χ2 test statistic used in three different circumstances. The following bulleted list is a summary that will help you decide which χ2 test is the appropriate one to use.
1. Goodness-of-Fit: Use the goodness-of-fit test to decide whether a population with an unknown distribution fits a known distribution. In this case there will be a single qualitative survey question or a single outcome of an experiment from a single population. Goodness-of-fit is typically used to see if the population is uniform (all outcomes occur with equal frequency), the population is normal, or the population is the same as another population with a known distribution. The null and alternative hypotheses are as follows:
H: The population fits the given distribution.
H: The population does not fit the given distribution.
2. Independence: Use the test for independence to decide whether two variables (factors) are independent or dependent. In this case there will be two qualitative survey questions or experiments and a contingency table will be constructed. The goal is to see if the two variables are unrelated/independent or related/dependent. The null and alternative hypotheses are as follows:
H: The two variables (factors) are independent.
H: The two variables (factors) are dependent.
3. Homogeneity: Use the test for homogeneity to decide if two populations with unknown distributions have the same distribution. In this case there will be a single qualitative survey question or experiment given to two different populations. The null and alternative hypotheses are as follows:
H: The two populations follow the same distribution.
H: The two populations have different distributions.
### Chapter Review
The goodness-of-fit test is typically used to determine if data fits a particular distribution. The test of independence makes use of a contingency table to determine the independence of two factors. The test for homogeneity determines whether two populations come from the same distribution, even if this distribution is unknown.
### Homework
For each word problem, use a solution sheet to solve the hypothesis test problem. Go to
Read the statement and decide whether it is true or false.
### Bringing It Together
|
# The Chi-Square Distribution
## Test of a Single Variance
A test of a single variance assumes that the underlying distribution is normal. The null and alternative hypotheses are stated in terms of the population variance or population standard deviation. The test statistic is
where
1. n = the total number of data,
2. s2 = sample variance, and
3. σ2 = population variance.
You may think of s as the random variable in this test. The number of degrees of freedom is df = n – 1. A test of a single variance may be right-tailed, left-tailed, or two-tailed. will show you how to set up the null and alternative hypotheses. The null and alternative hypotheses contain statements about the population variance.
### References
Apple Insider. (n.d.). Retrieved from http://appleinsider.com/mac_price_guide
World Bank. (n.d.). Retrieved from http://www.worldbank.org/
### Chapter Review
To test variability, use the chi-square test of a single variance. The test may be left-, right-, or two-tailed, and its hypotheses are always expressed in terms of the variance or standard deviation.
### Formula Review
Test of a single variance statistic where
n: sample size
s: sample standard deviation
σ: population standard deviation
df = n – 1 degrees of freedom
2. Use the test to determine variation.
3. The degrees of freedom is the number of samples – 1.
4. The test statistic is
, where n = the total number of data, s2 = sample variance, and σ2 = population variance.
5. The test may be left-, right-, or two-tailed.
Use the following information to answer the next three exercises. An archer’s standard deviation for his hits is six, where the data are measured in distance from the center of the target. An observer claims the standard deviation is less than six.
Use the following information to answer the next three exercises. The standard deviation of heights for students in a school is 0.81. A random sample of 50 students is taken, and the standard deviation of heights of the sample is 0.96. A researcher in charge of the study believes the standard deviation of heights for the school is greater than 0.81.
Use the following information to answer the next four exercises: The average waiting time in a doctor’s office varies. The standard deviation of waiting times in a doctor’s office is 3.4 minutes. A random sample of 30 patients in the doctor’s office has a standard deviation of waiting times of 4.1 minutes. One doctor believes the variance of waiting times is greater than originally thought.
### Homework
Use the following information to answer the next 12 exercises. Suppose an airline claims that its flights are consistently on time with an average delay of at most 15 minutes. It claims that the average delay is so consistent that the variance is no more than 150 minutes. Doubting the consistency part of the claim, a disgruntled traveler calculates the delays for his next 25 flights. The average delay for those 25 flights is 22 minutes with a standard deviation of 15 minutes.
For each word problem, use a solution sheet to solve the hypothesis test problem. Go to |
# Linear Regression and Correlation
## Introduction
Professionals often want to know how two or more numeric variables are related. For example, is there a relationship between the grade on the second math exam a student takes and the grade on the final exam? If there is a relationship, what is the relationship, and how strong is it?
In another example, your income may be determined by your education, your profession, your years of experience, and your ability. The amount you pay a repair person for labor is often determined by an initial amount plus an hourly fee.
The type of data described in the examples is bivariate data—bi—for two variables. In reality, statisticians use multivariate data, meaning many variables.
In this chapter, you will study the simplest form of regression—linear regression—with one independent variable (x). This involves data that fit a line in two dimensions. You will also study correlation, which measures the strength of a relationship. |
# Linear Regression and Correlation
## Linear Equations
Linear regression for two variables is based on a linear equation with one independent variable. The equation has the form
where a and b are constant numbers.
The variable x is the independent variable; y is the dependent variable. Typically, you choose a value to substitute for the independent variable and then solve for the dependent variable.
The graph of a linear equation of the form y = a + bx is a straight line. Any line that is not vertical can be described by this equation.
### Slope and y-interceptof a Linear Equation
For the linear equation y = a + bx, b = slope and a = y-inttercept. From algebra, recall that the slope is a number that describes the steepness of a line; the y-intercept is the y-coordinate of the point (0, a), where the line crosses the y-axis.
Please note that in previous courses you learned
was the slope-intercept form of the equation, where m represented the slope and b represented the y-intercept. In this text, the form
is used, where a is the y-intercept and b is the slope. The key is remembering the coefficient of x is the slope, and the constant number is the y-intercept.
### References
Centers for Disease Control and Prevention. (n.d.). Retrieved from https://www.cdc.gov/
National Center for HIV/AIDS, Viral Hepatitis, STD, and TB Prevention. (n.d.). Centers for Disease Control and Prevention. Retrieved from https://www.cdc.gov/nchhstp/default.htm
### Chapter Review
The most basic type of association is a linear association. This type of relationship can be defined algebraically by the equations used (numerically with actual or predicted data values) or graphically from a plotted curve. Lines are classified as straight curves. Algebraically, a linear equation typically takes the form y = mx + b, where m and b are constants, x is the independent variable, and y is the dependent variable. In a statistical context, a linear equation is written in the form y = a + bx, where a and b are the constants. This form is used to help you distinguish the statistical context from the algebraic context. In the equation y = a + bx, the constant b that multiplies the x variable (b is called a coefficient) is called the slope. The slope describes the rate of change between the independent and dependent variables; in other words, the rate of change describes the change that occurs in the dependent variable as the independent variable is changed. In the equation y = a + bx, the constant a is called the y-intercept. Graphically, the y-intercept is the y-coordinate of the point where the graph of the line crosses the y-axis. At this point, x = 0.
The slope of a line is a value that describes the rate of change between the independent and dependent variables. The slope tells us how the dependent variable (y) changes for every one-unit increase in the independent (x) variable, on average. The y-intercept is used to describe the dependent variable when the independent variable equals zero. Graphically, the slope is represented by three line types in elementary statistics.
### Formula Review
y = a + bx, where a is the y-intercept and b is the slope. The variable x is the independent variable and y is the dependent variable.
Use the following information to answer the next three exercises. A vacation resort rents scuba equipment to certified divers. The resort charges an up-front fee of $25 and another fee of $12.50 an hour.
Use the following information to answer the next two exercises. A credit card company charges $10 when a payment is late and $5 a day each day the payment remains unpaid.
Use the following information to answer the next exercise. contains real data for the first two decades of flu reporting.
Use the following information to answer the next two exercises. A specialty cleaning company charges an equipment fee and an hourly labor fee. A linear equation that expresses the total amount of the fee the company charges for each session is y = 50 + 100x.
Use the following information to answer the next three questions. As a result of erosion, a river shoreline is losing several thousand pounds of soil each year. A linear equation that expresses the total amount of soil lost per year is y = 12,000x.
Use the following information to answer the next two exercises. The price of a single issue of stock can fluctuate throughout the day. A linear equation that represents the price of stock for Shipment Express is y = 15 – 1.5x, where x is the number of hours passed in an eight-hour day of trading.
### Homework
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# Linear Regression and Correlation
## The Regression Equation
Data rarely fit a straight line exactly. Usually, you must be satisfied with rough
predictions. Typically, you have a set of data with a scatter plot that appear to fit a straight line. This is called a line of best fit or least-squares regression line.
The third exam score, x, is the independent variable, and the final exam score, y, is the dependent variable. We will plot a regression line that best fits the data. If each of you were to fit a line by eye, you would draw different lines. We can obtain a line of best fit using either the median-median line approach or by calculating the least-squares regression line.
Let's first find the line of best fit for the relationship between the third exam score and the final exam score using the median-median line approach. Remember that this is the data from after the ordered pairs have been listed by ordering x values. If multiple data points have the same y values, then they are listed in order from least to greatest y (see data values where x = 71). We first divide our scores into three groups of approximately equal numbers of x values per group. The first and third groups have the same number of x values. We must remember first to put the x values in ascending order. The corresponding y values are then recorded. However, to find the median, we first must rearrange the y values in each group from the least value to the greatest value. shows the correct ordering of the x values but does not show a reordering of the y values.
With this set of data, the first and last groups each have four x values and four corresponding y values. The second group has three x values and three corresponding y values. We need to organize the x and y values per group and find the median x and y values for each group. Let’s now write out our y values for each group in ascending order. For group 1, the y values in order are 126, 133, 153, and 175. For group 2, the y values are already in order. For group 3, the y values are also already in order. We can represent these data as shown in , but notice that we have broken the ordered pairs; (65, 126) is not a data point in our original set:
When this is completed, we can write the ordered pairs for the median values. This allows us to find the slope and y-intercept of the median-median line.
The ordered pairs are (66.5, 143), (69, 159), and (71, 174).
The slope can be calculated using the formula
Substituting the median x and y values from the first and third groups gives
which simplifies to
The y-intercept may be found using the formula
, which means the quantity of the sum of the median y values minus the slope times the sum of the median x values divided by three.
The sum of the median x values is 206.5, and the sum of the median y values is 476. Substituting these sums and the slope into the formula gives
, which simplifies to
The line of best fit is represented as
Thus, the equation can be written as y = 6.9x − 316.3.
The median–median line may also be found using your graphing calculator. You can enter the x and y values into two separate lists; choose Stat, Calc, Med-Med, and press Enter. The slope, a, and y-intercept, b, will be provided. The calculator shows a slight deviation from the previous manual calculation as a result of rounding. Rounding to the nearest tenth, the calculator gives the median-median line of
Each point of data is of the the form (x, y), and each point of the line of best fit using least-squares linear regression has the form (x, ŷ).
The ŷ is read y hat and is the estimated value of y. It is the value of y obtained using the regression line. It is not generally equal to y from data, but it is still important because it can help make predictions for other values.
The term y0 – ŷ0 = ε0 is called the error or residual. It is not an error in the sense of a mistake. The absolute value of a residual measures the vertical distance between the actual value of y and the estimated value of y. In other words, it measures the vertical distance between the actual data point and the predicted point on the line, or it measures how far the estimate is from the actual data value.
If the observed data point lies above the line, the residual is positive and the line underestimates the actual data value for y. If the observed data point lies below the line, the residual is negative and the line overestimates that actual data value for y.
In , y0 – ŷ0 = ε0 is the residual for the point shown. Here the point lies above the line and the residual is positive.
ε = the Greek letter epsilon
For each data point, you can calculate the residuals or errors, yi ŷi = εi for i = 1, 2, 3, . . . , 11.
Each |ε| is a vertical distance.
For the example about the third exam scores and the final exam scores for the 11 statistics students, there are 11 data points. Therefore, there are 11 ε values. If you square each ε and add them, you get the sum of ε squared from i = 1 to i = 11, as shown below.
This is called the sum of squared errors (SSE).
Using calculus, you can determine the values of a and b that make the SSE a minimum. When you make the SSE a minimum, you have determined the points that are on the line of best fit. It turns out that the line of best fit has the equation
where
and
.
The sample means of the x values and the y values are
and
, respectively. The best-fit line always passes through the point
.
The slope (b) can be written as
where s = the standard deviation of the y values and s = the standard deviation of the x values. r is the correlation coefficient, which shows the relationship between the x and y values. This will be discussed in more detail in the next section.
### Least-Squares Criteria for Best Fit
The process of fitting the best-fit line is called linear regression. We assume that the data are scattered about a straight line. To find that line, we minimize the sum of the squared errors (SSE), or make it as small as possible. Any other line you might choose would have a higher SSE than the best-fit line. This best-fit line is called the least-squares regression line.
### Third Exam vs. Final Exam Example
The graph of the line of best fit for the third exam/final exam example is as follows:
The least-squares regression line (best-fit line) for the third exam/final exam example has the equation
### Understanding and Interpreting the y-intercept
The y-intercept, a, of the line describes where the plot line crosses the y-axis. The y-intercept of the best-fit line tells us the best value of the relationship when x is zero. In some cases, it does not make sense to figure out what y is when x = 0. For example, in the third exam vs. final exam example, the y-intercept occurs when the third exam score, or x, is zero. Since all the scores are grouped around a passing grade, there is no need to figure out what the final exam score, or y, would be when the third exam was zero.
However, the y-intercept is very useful in many cases. For many examples in science, the y-intercept gives the baseline reading when the experimental conditions aren't applied to an experimental system. This baseline indicates how much the experimental condition affects the system. It could also be used to ensure that equipment and measurements are calibrated properly before starting the experiment.
In biology, the concentration of proteins in a sample can be measured using a chemical assay that changes color depending on how much protein is present. The more protein present, the darker the color. The amount of color can be measured by the absorbance reading. shows the expected absorbance readings at different protein concentrations. This is called a standard curve for the assay.
The scatter plot includes the line of best fit.
The y-intercept of this line occurs at 0.0226 mAU. This means the assay gives a reading of 0.0226 mAU when there is no protein present. That is, it is the baseline reading that can be attributed to something else, which, in this case, is some other non-protein chemicals that are absorbing light. We can tell that this line of best fit is reasonable because the y-intercept is small, close to zero. When there is no protein present in the sample, we expect the absorbance to be very small, or close to zero, as well.
### Understanding Slope
The slope of the line, b, describes how changes in the variables are related. It is important to interpret the slope of the line in the context of the situation represented by the data. You should be able to write a sentence interpreting the slope in plain English.
Interpretation of the Slope: The slope of the best-fit line tells us how the dependent variable (y) changes for every one unit increase in the independent (x) variable, on average.
Third Exam vs. Final Exam ExampleSlope: The slope of the line is b = 4.83.
Interpretation: For a 1-point increase in the score on the third exam, the final exam score increases by 4.83 points, on average.
### The Correlation Coefficient r
Besides looking at the scatter plot and seeing that a line seems reasonable, how can you determine whether the line is a good predictor? Use the correlation coefficient as another indicator (besides the scatter plot) of the strength of the relationship between x and y.
The correlation coefficient, developed by Karl Pearson during the early 1900s, is numeric and provides a measure of the strength and direction of the linear association between the independent variable x and the dependent variable y.
If you suspect a linear relationship between x and y, then r can measure the strength of the linear relationship.
What the Value of
What the Sign of
The correlation coefficient is calculated as the quantity of data points times the sum of the quantity of the x-coordinates times the y-coordinates, minus the quantity of the sum of the x-coordinates times the sum of the y-coordinates, all divided by the square root of the quantity of data points times the sum of the x-coordinates squared minus the square of the sum of the x-coordinates, times the number of data points times the sum of the y-coordinates squared minus the square of the sum of the y-coordinates. It can be summarized by the following equation:
where n is the number of data points.
The formula for r looks formidable. However, computer spreadsheets, statistical software, and many calculators can calculate r quickly. The correlation coefficient, r, is the bottom item in the output screens for the LinRegTTest on the TI-83, TI-83+, or TI-84+ calculator (see previous section for instructions).
### The Coefficient of Determination
The variable r2 is called the coefficient of determination and it is the square of the correlation coefficient, but it is usually stated as a percentage, rather than in decimal form. It has an interpretation in the context of the data:
1.
when expressed as a percent, represents the percentage of variation in the dependent (predicted) variable y that can be explained by variation in the independent (explanatory) variable x using the regression (best-fit) line.
2. 1 – when expressed as a percentage, represents the percentage of variation in y that is not explained by variation in x using the regression line. This can be seen as the scattering of the observed data points about the regression line.
Consider the third exam/final exam example introduced in the previous section.
Interpret r2 in the context of this example.
### Chapter Review
A regression line, or a line of best fit, can be drawn on a scatter plot and used to predict outcomes for the x and y variables in a given data set or sample data. There are several ways to find a regression line, but usually the least-squares regression line is used because it creates a uniform line. Residuals, also called errors, measure the distance from the actual value of y and the estimated value of y. The sum of squared errors, or SSE, when set to its minimum, calculates the points on the line of best fit. Regression lines can be used to predict values within the given set of data but should not be used to make predictions for values outside the set of data.
The correlation coefficient, r, measures the strength of the linear association between x and y. The variable r has to be between –1 and +1. When r is positive, x and y tend to increase and decrease together. When r is negative, x increases and y decreases, or the opposite occurs: x decreases and y increases. The coefficient of determination, r2, is equal to the square of the correlation coefficient. When expressed as a percentage, r2 represents the percentage of variation in the dependent variable, y, that can be explained by variation in the independent variable, x, using the regression line.
Use the following information to answer the next five exercises. A random sample of 10 professional athletes produced the following data, where x is the number of endorsements the player has and y is the amount of money made, in millions of dollars.
### Homework
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# Linear Regression and Correlation
## Testing the Significance of the Correlation Coefficient (Optional)
The correlation coefficient, r, tells us about the strength and direction of the linear relationship between x and y. However, the reliability of the linear model also depends on how many observed data points are in the sample. We need to look at both the correlation coefficient r and the sample size n, together.
We perform a hypothesis test of the significance of the correlation coefficient to decide whether the linear relationship in the sample data is strong enough to use to model the relationship in the population.
The sample data are used to compute r, the correlation coefficient for the sample. If we had data for the entire population, we could find the population correlation coefficient. But, because we have only sample data, we cannot calculate the population correlation coefficient. The sample correlation coefficient, r, is our estimate of the unknown population correlation coefficient.
1. The symbol for the population correlation coefficient is ρ, the Greek letter rho.
2. ρ = population correlation coefficient (unknown).
3. r = sample correlation coefficient (known; calculated from sample data).
The hypothesis test lets us decide whether the value of the population correlation coefficient ρ is close to zero or significantly different from zero. We decide this based on the sample correlation coefficient r and the sample size n.
If the test concludes the correlation coefficient is significantly different from zero, we say the correlation coefficient is significant.
If the test concludes the correlation coefficient is not significantly different from zero (it is close to zero), we say the correlation coefficient is not significant.
### Performing the Hypothesis Test
1. Null hypothesis: H0: ρ = 0.
2. Alternate hypothesis: H: ρ ≠ 0.
What the Hypothesis Means in Words:
1. Null hypothesis The population correlation coefficient is not significantly different from zero. There is not a significant linear relationship (correlation) between x and y in the population.
2. Alternate hypothesis The population correlation coefficient is significantly different from zero. There is a significant linear relationship (correlation) between x and y in the population.
Drawing a Conclusion:There are two methods to make a conclusion. The two methods are equivalent and give the same result.
1. Method 1: Use the
2. Method 2: Use a table of critical values.
In this chapter, we will always use a significance level of 5 percent, α = 0.05.
### METHOD 1: Using a p-value to Make a Decision
2. Decision: Reject the null hypothesis.
3. Conclusion: There is sufficient evidence to conclude there is a significant linear relationship between x and y because the correlation coefficient is significantly different from zero.
2. Decision: Do not reject the null hypothesis.
3. Conclusion: There is insufficient evidence to conclude there is a significant linear relationship between x and y because the correlation coefficient is not significantly different from zero.
You will use technology to calculate the p-value, but it is useful to know that the p-value is calculated using a t distribution with n – 2 degrees of freedom and that the p-value is the combined area in both tails.
An alternative way to calculate the p-value (p) given by LinRegTTest is the command 2*tcdf(abs(t),10^99, n–2) in 2nd DISTR.
2. Consider the third exam/final exam example.
3. The line of best fit is ŷ = 173.51 + 4.83x, with r = 0.6631, and there are n = 11 data points.
4. Can the regression line be used for prediction? Given a third exam score (x value), can we
use the line to predict the final exam score (predicted y value)?
H0: ρ = 0
H: ρ ≠ 0
α = 0.05
1. The p-value is 0.026 (from LinRegTTest on a calculator or from computer software).
2. The p-value, 0.026, is less than the significance level of α = 0.05.
3. Decision: Reject the null hypothesis H0.
4. Conclusion: There is sufficient evidence to conclude there is a significant linear relationship between the third exam score (x) and the final exam score (y) because the correlation coefficient is significantly different from zero.
Because r is significant and the scatter plot shows a linear trend, the regression line can be used to predict final exam scores.
### METHOD 2: Using a Table of Critical Values to Make a Decision
The 95 Percent Critical Values of the Sample Correlation Coefficient Table () can be used to give you a good idea of whether the computed value of r is significant. Use it to find the critical values using the degrees of freedom, df = n – 2. The table has already been calculated with α = 0.05. The table tells you the positive critical value, but you should also make that number negative to have two critical values. If r is not between the positive and negative critical values, then the correlation coefficient is significant. If r is significant, then you may use the line for prediction. If r is not significant (between the critical values), you should not use the line to make predictions.
### Third Exam vs. Final Exam Example: Critical Value Method
Consider the third exam/final exam example.
The line of best fit is: ŷ = –173.51 + 4.83x, with r = .6631, and there are n = 11 data points. Can the regression line be used for prediction? Given a third exam score (x value), can we use the line to predict the final exam score (predicted y value)?
1. H0: ρ = 0
2. H: ρ ≠ 0
3. α = 0.05
1. Use the 95 Percent Critical Values table for r with df = n – 2 = 11 – 2 = 9.
2. Using the table with df = 9, we find that the critical value listed is 0.602. Therefore, the critical values are ±0.602.
3. Since 0.6631 > 0.602, r is significant.
4. Decision: Reject the null hypothesis.
5. Conclusion: There is sufficient evidence to conclude there is a significant linear relationship between the third exam score (x) and the final exam score (y) because the correlation coefficient is significantly different from zero.
Because r is significant and the scatter plot shows a linear trend, the regression line can be used to predict final exam scores.
### Assumptions in Testing the Significance of the Correlation Coefficient
Testing the significance of the correlation coefficient requires that certain assumptions about the data be satisfied. The premise of this test is that the data are a sample of observed points taken from a larger population. We have not examined the entire population because it is not possible or feasible to do so. We are examining the sample to draw a conclusion about whether the linear relationship that we see between x and y in the sample data provides strong enough evidence that we can conclude there is a linear relationship between x and y in the population.
The regression line equation that we calculate from the sample data gives the best-fit line for our particular sample. We want to use this best-fit line for the sample as an estimate of the best-fit line for the population. Examining the scatter plot and testing the significance of the correlation coefficient helps us determine whether it is appropriate to do this.
2. There is a linear relationship in the population that models the sample data. Our regression line from the sample is our best estimate of this line in the population.
3. The y values for any particular x value are normally distributed about the line. This implies there are more y values scattered closer to the line than are scattered farther away. Assumption 1 implies that these normal distributions are centered on the line; the means of these normal distributions of y values lie on the line.
4. Normal distributions of all the y values have the same shape and spread about the line.
5. The residual errors are mutually independent (no pattern).
6. The data are produced from a well-designed, random sample or randomized experiment.
### Chapter Review
Linear regression is a procedure for fitting a straight line of the form ŷ = a + bx to data. The conditions for regression are as follows:
1. Linear: In the population, there is a linear relationship that models the average value of y for different values of x.
2. Independent: The residuals are assumed to be independent.
3. Normal: The y values are distributed normally for any value of x.
4. Equal variance: The standard deviation of the y values is equal for each x value.
5. Random: The data are produced from a well-designed random sample or a randomized experiment.
The slope b and intercept a of the least-squares line estimate the slope β and intercept α of the population (true) regression line. To estimate the population standard deviation of y (σ) use the standard deviation of the residuals:
. The variable ρ (rho) is the population correlation coefficient. To test the null hypothesis, H0: ρ = hypothesized value, use a linear regression t-test. The most common null hypothesis is H0: ρ = 0, which indicates there is no linear relationship between x and y in the population. The TI-83, 83+, 84, 84+ calculator function LinRegTTest can perform this test (STATS, TESTS, LinRegTTest).
### Formula Review
Least-Squares Line or Line of Best Fit:
where a is the y-intercept and b is the slope.
Standard Deviation of the Residuals:
where SSE = sum of squared errors, and
n = the number of data points. |
# Linear Regression and Correlation
## Prediction (Optional)
Recall the third exam/final exam example.
We found the equation of the best-fit line for the final exam grade as a function of the grade on the third exam. We can now use the least-squares regression line for prediction.
Suppose you want to estimate, or predict, the mean final exam score of statistics students who received a 73 on the third exam. The exam scores (x values) range from 65 to 75. Since 73 is between the x values 65 and 75, substitute x = 73 into the equation. Then,
We predict that statistics students who earn a grade of 73 on the third exam will earn a grade of 179.08 on the final exam, on average.
### References
Centers for Disease Control and Prevention. (n.d.). Retrieved from https://www.cdc.gov/
National Center for HIV/AIDS, Viral Hepatitis, STD, and TB Prevention. (n.d.). Centers for Disease Control and Prevention. Retrieved from https://www.cdc.gov/nchhstp/default.htm
National Center for Health Statistics. (n.d.). Centers for Disease Control and Prevention. Retrieved from https://www.cdc.gov/nchs/index.htm
U.S. Census Bureau. (n.d.). Retrieved from http://www.census.gov/compendia/statab/cats/transportation/motor_vehicle_accidents_and_fatalities.html
### Chapter Review
After determining the presence of a strong correlation coefficient and calculating the line of best fit, you can use the least-squares regression line to make predictions about your data.
Use the following information to answer the next two exercises. An electronics retailer used regression to find a simple model to predict sales growth in the first quarter of the new year (January through March). The model is good for 90 days, where x is the day. The model can be written as ŷ = 101.32 + 2.48x, where ŷ is in thousands of dollars.
Use the following information to answer the next three exercises. A landscaping company is hired to mow the grass for several large properties. The total area of the properties is 1,345 acres. The rate at which one person can mow is ŷ = 1350 – 1.2x, where x is the number of hours and ŷ represents the number of acres left to mow.
Use the following information to answer the next 14 exercises. contains real data for the first two decades of flu reporting.
### Homework
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# Linear Regression and Correlation
## Outliers
In some data sets, there are values (observed data points) called outliers. Outliers are observed data points that are far from the least-squares line. They have large errors, where the error or residual is not very close to the best-fit line.
Outliers need to be examined closely. Sometimes, they should not be included in the analysis of the data, like if it is possible that an outlier is a result of incorrect data. Other times, an outlier may hold valuable information about the population under study and should remain included in the data. The key is to examine carefully what causes a data point to be an outlier.
Besides outliers, a sample may contain one or a few points that are called influential points. Influential points are observed data points that are far from the other observed data points in the horizontal direction. These points may have a big effect on the slope of the regression line. To begin to identify an influential point, you can remove it from the data set and determine whether the slope of the regression line is changed significantly.
You also want to examine how the correlation coefficient, r, has changed. Sometimes, it is difficult to discern a significant change in slope, so you need to look at how the strength of the linear relationship has changed. Computers and many calculators can be used to identify outliers and influential points. Regression analysis can determine if an outlier is, indeed, an influential point. The new regression will show how omitting the outlier will affect the correlation among the variables, as well as the fit of the line. A graph showing both regression lines helps determine how removing an outlier affects the fit of the model.
### Identifying Outliers
We could guess at outliers by looking at a graph of the scatter plot and best-fit line. However, we would like some guideline regarding how far away a point needs to be to be considered an outlier. As a rough rule of thumb, we can flag as an outlier any point that is located farther than two standard deviations above or below the best-fit line. The standard deviation used is the standard deviation of the residuals or errors.
We can do this visually in the scatter plot by drawing an extra pair of lines that are two standard deviations above and below the best-fit line. Any data points outside this extra pair of lines are flagged as potential outliers. Or, we can do this numerically by calculating each residual and comparing it with twice the standard deviation. With regard to the TI-83, 83+, or 84+ calculators, the graphical approach is easier. The graphical procedure is shown first, followed by the numerical calculations. You would generally need to use only one of these methods.
### Numerical Identification of Outliers
In , the first two columns include the third exam and final exam data. The third column shows the predicted ŷ values calculated from the line of best fit: ŷ = –173.5 + 4.83x. The residuals, or errors, that were mentioned in Section 3 of this chapter have been calculated in the fourth column of the table: Observed y value predicted y value = y ŷ.
s is the standard deviation of all the y ŷ = ε values, where n is the total number of data points. If each residual is calculated and squared, and the results are added, we get the SSE. The standard deviation of the residuals is calculated from the SSE as
Rather than calculate the value of s ourselves, we can find s using a computer or calculator. For this example, the calculator function LinRegTTest found s = 16.4 as the standard deviation of the residuals .
We are looking for all data points for which the residual is greater than 2s = 2(16.4) = 32.8 or less than –32.8. Compare these values with the residuals in column four of the table. The only such data point is the student who had a grade of 65 on the third exam and 175 on the final exam; the residual for this student is 35.
### How Does the Outlier Affect the Best-Fit Line?
Numerically and graphically, we have identified point (65, 175) as an outlier. Recall that recalculation of the least-squares regression line and summary statistics, following deletion of an outlier, may be used to determine whether an outlier is also an influential point. This process also allows you to compare the strength of the correlation of the variables and possible changes in the slope both before and after the omission of any outliers.
Compute a new best-fit line and correlation coefficient using the 10 remaining points.
On the TI-83, TI-83+, or TI-84+ calculators, delete the outlier from L1 and L2. Using the LinRegTTest, found under Stat and Tests, the new line of best fit and correlation coefficient are the following:
and
.
The slope is now 7.39, compared to the previous slope of 4.83. This seems significant, but we need to look at the change in r-values as well. The new line shows
, which indicates a stronger correlation than the original line, with
since
is closer to 1. This means the new line is a better fit to the data values. The line can better predict the final exam score given the third exam score. It also means the outlier of (65, 175) was an influential point, since there is a sizeable difference in r-values. We must now decide whether to delete the outlier. If the outlier was recorded erroneously, it should certainly be deleted. Because it produces such a profound effect on the correlation, the new line of best fit allows for better prediction and an overall stronger model.
You may use Excel to graph the two least-squares regression lines and compare the slopes and fit of the lines to the data, as shown in .
You can see that the second graph shows less deviation from the line of best fit. It is clear that omission of the influential point produced a line of best fit that more closely models the data.
### Numerical Identification of Outliers: Calculating s and Finding Outliers Manually
If you do not have the function LinRegTTest on your calculator, then you must calculate the outlier in the first example by doing the following.
First, square each |y – ŷ|.
The squares are
Then, add (sum) all the |y – ŷ| squared terms using the formula
(Recall that yi – ŷi = εi).
= 352 + 172 + 162 + 62 + 192 + 92 + 32 + 12 + 102 + 92 + 12
= 2,440 = SSE.
The result, SSE, is the sum of squared errors.
Next, calculate s, the standard deviation of all the y – ŷ = ε-values where n = the total number of data points.
The calculation is .
For the third exam/final exam example,
Next, multiply s by 2:
(2)(16.47) = 32.94
32.94 is two standard deviations away from the mean of the y – ŷ values.
If we were to measure the vertical distance from any data point to the corresponding point on the line of best fit and that distance is at least 2s, then we would consider the data point to be too far from the line of best fit. We call that point a potential outlier.
For the example, if any of the |y – ŷ| values are at least 32.94, the corresponding (x, y) data point is a potential outlier.
For the third exam/final exam example, all the |y – ŷ| values are less than 31.29 except for the first one, which is 35:
35 > 31.29. That is, |y – ŷ| ≥ (2)(s).
The point that corresponds to |y – ŷ| = 35 is (65, 175). Therefore, the data point (65, 175) is a potential outlier. For this example, we will delete it. (Remember, we do not always delete an outlier.)
The next step is to compute a new best-fit line using the 10 remaining points. The new line of best fit and the correlation coefficient are
ŷ = –355.19 + 7.39x and r = .9121.
### 95 Percent Critical Values of the Sample Correlation Coefficient Table
### References
Committee on Ways and Means, U.S. House of Representatives. (n.d.). Washington, DC: U.S. Department of Health and Human Services.
Microsoft Bookshelf. (n.d.).
Physicians Desk Reference Staff. (1990). Physicians desk reference. Ohio: Medical Economics Company.
U.S. Bureau of Labor Statistics. (n.d.). Retrieved from https://www.bls.gov/
### Chapter Review
To determine whether a point is an outlier, do one of the following:
Use the following information to answer the next four exercises. The scatter plot shows the relationship between hours spent studying and exam scores. The line shown is the calculated line of best fit. The correlation coefficient is 0.69.
### Homework
### Bring It Together
Use the following information to answer the next two exercises. The cost of a leading liquid laundry detergent in different sizes is given in . |
# F Distribution and One-way Anova
## Introduction
Many statistical applications in psychology, social science, business administration, and the natural sciences involve several groups. For example, an environmentalist is interested in knowing if the average amount of pollution varies among several bodies of water. A sociologist is interested in knowing if the amount of income a person earns varies according to his or her upbringing. A consumer looking for a new car might compare the average gas mileage of several models.
For hypothesis tests comparing averages across more than two groups, statisticians have developed a method called analysis of variance (abbreviated ANOVA). In this chapter, you will study the simplest form of ANOVA called single factor or one-way ANOVA. You will also study the F distribution, used for one-way ANOVA, and the test of two variances. This is a very brief overview of one-way ANOVA. You will study this topic in much greater detail in future statistics courses. One-way ANOVA, as it is presented here, relies heavily on a calculator or computer. |
# F Distribution and One-way Anova
## One-Way ANOVA
The purpose of a one-way ANOVA test is to determine the existence of a statistically significant difference among several group means. The test uses variances to help determine if the means are equal or not. To perform a one-way ANOVA test, there are five basic assumptions to be fulfilled:
1. Each population from which a sample is taken is assumed to be normal.
2. All samples are randomly selected and independent.
3. The populations are assumed to have equal standard deviations (or variances).
4. The factor is a categorical variable.
5. The response is a numerical variable.
### The Null and Alternative Hypotheses
The null hypothesis is that all the group population means are the same. The alternative hypothesis is that at least one pair of means is different. For example, if there are k groups
H: μ1 = μ2 = μ3 = ... = μ
H: At least two of the group means μ1, μ2, μ3, ..., μ are not equal. That is, μ ≠ μ for some i ≠ j.
The graphs, a set of box plots representing the distribution of values with the group means indicated by a horizontal line through the box, help in the understanding of the hypothesis test. In the first graph (red box plots), H: μ = μ = μ and the three populations have the same distribution if the null hypothesis is true. The variance of the combined data is approximately the same as the variance of each of the populations.
If the null hypothesis is false, then the variance of the combined data is larger, which is caused by the different means as shown in the second graph (green box plots).
### Chapter Review
Analysis of variance extends the comparison of two groups to several, each a level of a categorical variable (factor). Samples from each group are independent and must be randomly selected from normal populations with equal variances. We test the null hypothesis of equal means of the response in every group versus the alternative hypothesis of one or more group means being different from the others. A one-way ANOVA hypothesis test determines if several population means are equal. The distribution for the test is the F distribution with two different degrees of freedom.
2. Each population from which a sample is taken is assumed to be normal.
3. All samples are randomly selected and independent.
4. The populations are assumed to have equal standard deviations (or variances).
Use the following information to answer the next five exercises. There are five basic assumptions that must be fulfilled to perform a one-way ANOVA test. What are they?
### Homework
|
# F Distribution and One-way Anova
## The F Distribution and the F Ratio
The distribution used for the hypothesis test is a new one. It is called the , named after Sir Ronald Fisher, an English statistician. The F statistic is a ratio (a fraction). There are two sets of degrees of freedom: one for the numerator and one for the denominator.
For example, if F follows an F distribution and the number of degrees of freedom for the numerator is 4, and the number of degrees of freedom for the denominator is 10, then F ~ F.
To calculate the , two estimates of the variance are made.
1. Variance between samples: an estimate of σ2 that is the variance of the sample means multiplied by n, when the sample sizes are the same. If the samples are different sizes, the variance between samples is weighted to account for the different sample sizes. The variance is also called variation due to treatment or explained variation.
2. Variance within samples: an estimate of σ2 that is the average of the sample variances, also known as a pooled variance. When the sample sizes are different, the variance within samples is weighted. The variance is also called the variation due to error or unexplained variation.
1. SSbetween = the sum of squares that represents the variation among the different samples
2. SSwithin = the sum of squares that represents the variation within samples that is due to chance
To find a sum of squares mean, add together squared quantities which, in some
cases, may be weighted. We used sum of squares to calculate the sample variance and the sample standard deviation in Descriptive Statistics.
MS means mean square. MSbetween is the variance between groups, and MSwithin is the variance within groups.
2. k = the number of different groups
3. n = the size of the j group
4. s = the sum of the values in the j group
5. n = total number of all the values combined (total sample size: ∑n)
6. x = one value: ∑x = ∑s
7. Sum of squares of all values from every group combined: ∑x2
8. Between group variability: SStotal = ∑x2 –
9. Total sum of squares: ∑x2 –
10. Explained variation: sum of squares representing variation among the different samples
11. Unexplained variation: sum of squares representing variation within samples due to chance
12. dfs for different groups (dfs for the numerator): df = k – 1
13. Equation for errors within samples (dfs for the denominator): dfwithin = n – k
14. Mean square (variance estimate) explained by the different groups: MSbetween =
15. Mean square (variance estimate) that is due to chance (unexplained): MSwithin =
MSbetween and MSwithin can be written as follows:
1.
2.
The one-way ANOVA test depends on the fact that MSbetween can be influenced by population differences among means of the several groups. Since MSwithin compares values of each group to its own group mean, the fact that group means might be different does not affect MSwithin.
The null hypothesis says that all groups are samples from populations having the same normal distribution. The alternate hypothesis says that at least two of the sample groups come from populations with different normal distributions. If the null hypothesis is true, MSbetween and MSwithin should both estimate the same value.
If MSbetween and MSwithin estimate the same value, following the belief that H is true, then the F ratio should be approximately equal to 1. Mostly, just sampling errors would contribute to variations away from 1. As it turns out, MSbetween consists of the population variance plus a variance produced from the differences between the samples. MSwithin is an estimate of the population variance. Since variances are always positive, if the null hypothesis is false, MSbetween will generally be larger than MSwithin. Then the F ratio will be larger than 1. However, if the population effect is small, it is not unlikely that MSwithin will be larger in a given sample.
The previous calculations were done with groups of different sizes. If the groups are the same size, the calculations simplify somewhat and the F ratio can be written as follows:
2. n = the sample size
3. dfnumerator = k – 1
4. dfdenominator = n – k
5. s2 pooled = the mean of the sample variances (pooled variance)
6.
= the variance of the sample means
Data is typically put into a table for easy viewing. One-way ANOVA results are often displayed in this manner by computer software.
The one-way ANOVA hypothesis test is always right-tailed because larger F values are way out in the right tail of the F distribution curve and tend to make us reject H.
### Notation
The notation for the F distribution is F ~ F,
where df(num) = dfbetween and df(denom) = dfwithin.
The mean for the F distribution is
### References
Marist College School of Science. (n.d.). Tomato data (Unpublished student research). Marist College School of Science, Poughkeepsie, NY.
### Chapter Review
Analysis of variance compares the means of a response variable for several groups. ANOVA compares the variation within each group to the variation of the mean of each group. The ratio of these two is the F statistic from an F distribution with (number of groups – 1) as the numerator degrees of freedom and (number of observations – number of groups) as the denominator degrees of freedom. These statistics are summarized in the ANOVA table.
### Formula Review
dfbetween = df(num) = k – 1
dfwithin = df(denom) = n – k
MSbetween =
MSwithin =
F =
F ratio when the groups are the same size: F =
Mean of the F distribution: µ =
where
Use the following information to answer the next seven exercises. Groups of men from three different areas of the country are to be tested for mean weight. The entries in are the weights for the different groups.
Use the following information to answer the next eight exercises. Girls from four different soccer teams are to be tested for mean goals scored per game. The entries in are the goals per game for the different teams.
### Homework
Use the following information to answer the next three exercises. Suppose a group is interested in determining whether teenagers obtain their drivers licenses at approximately the same average age across the country.
Suppose that the following data are randomly collected from five teenagers in each region of the country. The numbers represent the age at which teenagers obtained their drivers licenses.
H: µ1 = µ2 = µ3 = µ4 = µ5
Hα: At least any two of the group means µ1, µ2, …, µ5 are not equal. |
# F Distribution and One-way Anova
## Facts About the F Distribution
The following are facts about the F distribution:
1. The curve is not symmetrical but skewed to the right.
2. There is a different curve for each set of dfs.
3. The F statistic is greater than or equal to zero.
4. As the degrees of freedom for the numerator and for the denominator get larger, the curve approximates the normal.
5. Other uses for the F distribution include comparing two variances and two-way analysis of variance. Two-way analysis is beyond the scope of this chapter.
### References
ESPN. (2012). MLB standings 2012. Retrieved from http://espn.go.com/mlb/standings/_/year/2012.
Hand, D. J. et al. (1994). A Handbook of Small Datasets: Data for Fruitfly Fecundity. London: Chapman & Hall.
Hand, D. J. et al. (1994). A Handbook of Small Datasets. London: Chapman & Hall, p. 50.
Hand. A Handbook of Small Datasets. p. 118.
Mackowiak, P. A.,Wasserman, S. S., & Levine, M. M. (1992). A critical appraisal of 98.6 degrees F, the upper limit of the normal body temperature, and other legacies of Carl Reinhold August Wunderlich. Journal of the American Medical Association, 268, 15781580.
Private K12 school in San Jose, CA. (1994). Data from a fourth grade classroom.
### Chapter Review
The graph of the F distribution is always positive and skewed right, though the shape can be mounded or exponential depending on the combination of numerator and denominator degrees of freedom. The F statistic is the ratio of a measure of the variation in the group means to a similar measure of the variation within the groups. If the null hypothesis is correct, then the numerator should be small compared to the denominator. A small F statistic will result, and the area under the F curve to the right will be large, representing a large p-value. When the null hypothesis of equal group means is incorrect, then the numerator should be large compared to the denominator, giving a large F statistic and a small area (small p-value) to the right of the statistic under the F curve.
When the data have unequal group sizes (unbalanced data), then techniques from The F Distribution and the F Ratio need to be used for hand calculations. In the case of balanced data, where the groups are the same size, simplified calculations based on group means and variances may be used. In practice, software is usually employed in the analysis. As in any analysis, graphs of various sorts should be used in conjunction with numerical techniques. Always look at your data!
Use the following information to answer the next seven exercises. Four basketball teams took a random sample of players regarding how high each player can jump (in inches). The results are shown in .
Use the following information to answer the next seven exercises. A video game developer is testing a new game on three different groups. Each group represents a different target market for the game. The developer collects scores from a random sample from each group. The results are shown in .
Use the following information to answer the next three exercises. Suppose a group is interested in determining whether teenagers obtain their drivers licenses at approximately the same average age across the country. Suppose that the following data are randomly collected from five teenagers in each region of the country. The numbers represent the age at which teenagers obtained their drivers licenses.
Enter the data into your calculator or computer.
State the decisions and conclusions (in complete sentences) for the following preconceived levels of α.
### Homework
Use the following information to answer the next two exercises. lists the number of pages in four different types of magazines. |
# F Distribution and One-way Anova
## Test of Two Variances
Another use of the F distribution is testing two variances. It is often desirable to compare two variances rather than two averages. For instance, college administrators would like two college professors grading exams to have the same variation in their grading. For a lid to fit a container, the variation in the lid and the container should be the same. A supermarket might be interested in the variability of check-out times for two checkers.
To perform a F test of two variances, it is important that the following are true:
Unlike most other tests in this book, the F test for equality of two variances is very sensitive to deviations from normality. If the two distributions are not normal, the test can give higher p-values than it should, or lower ones, in ways that are unpredictable. Many texts suggest that students not use this test at all, but in the interest of completeness we include it here.
Suppose we sample randomly from two independent normal populations. Let
and
be the population variances and
and
be the sample variances. Let the
sample sizes be n1 and n2. Since we are interested in comparing the two sample variances, we use the F ratio
F has the distribution F ~ F(n1 – 1, n2 – 1),
where n1 – 1 are the degrees of freedom for the numerator and n2 – 1 are the degrees of freedom for the denominator.
If the null hypothesis is
, then the F ratio becomes
.
If the two populations have equal variances, then
and
are close in value and
is close to 1. But if the two population variances are very different,
and
tend to be very different, too. Choosing
as the larger sample variance causes the ratio
to be greater than 1. If
and
are far apart, then
is a large number.
Therefore, if F is close to 1, the evidence favors the null hypothesis (the two population variances are equal). But if F is much larger than 1, then the evidence is against the null hypothesis. A test of two variances may be left-tailed, right-tailed, or two-tailed.
### References
ESPN. (2012). MLB standings 2012. Retrieved from http://espn.go.com/mlb/standings/_/year/2012/type/vs-division/order/true.
### Chapter Review
The F test for the equality of two variances rests heavily on the assumption of normal distributions. The test is unreliable if this assumption is not met. If both distributions are normal, then the ratio of the two sample variances is distributed as an F statistic, with numerator and denominator degrees of freedom that are one less than the samples sizes of the corresponding two groups. A test of two variances hypothesis test determines if two variances are the same. The distribution for the hypothesis test is the F distribution with two different degrees of freedom.
2. The populations from which the two samples are drawn are normally distributed.
3. The two populations are independent of each other.
### Formula Review
F has the distribution F ~ F(n1 – 1, n2 – 1)
F =
If σ1 = σ2, then F =
Use the following information to answer the next two exercises. There are two assumptions that must be true to perform an F test of two variances.
Use the following information to answer the next seven exercises. Two coworkers commute from the same building. They are interested in whether there is any variation in the time it takes them to drive to work. They each record their times for 20 commutes. The first worker’s times have a variance of 12.1. The second worker’s times have a variance of 16.9. The first worker thinks that he is more consistent with his commute times. Test the claim at the 10 percent level. Assume that commute times are normally distributed.
Use the following information to answer the next four exercises. Two students are interested in whether there is variation in their test scores for math class. There are 15 total math tests they have taken so far. The first student’s grades have a standard deviation of 38.1. The second student’s grades have a standard deviation of 22.5. The second student thinks his scores are more consistent.
Use the following information to answer the next three exercises. Two cyclists are comparing the variances of their overall paces going uphill. Each cyclist records his or her speeds going up 35 hills. The first cyclist has a variance of 23.8, and the second cyclist has a variance of 32.1. The cyclists want to see if their variances are the same or different. Assume that speeds are normally distributed.
### Homework
Use the following information to answer the next two exercises. The following table lists the number of pages in four different types of magazines. |
# What is Physics?
## Introduction
Take a look at the image above of the Andromeda Galaxy (), which contains billions of stars. This galaxy is the nearest one to our own galaxy (the Milky Way) but is still a staggering 2.5 million light years from Earth. (A light year is a measurement of the distance light travels in a year.) Yet, the primary force that affects the movement of stars within Andromeda is the same force that we contend with here on Earth—namely, gravity.
You may soon realize that physics plays a much larger role in your life than you thought. This section introduces you to the realm of physics, and discusses applications of physics in other disciplines of study. It also describes the methods by which science is done, and how scientists communicate their results to each other. |
# What is Physics?
## Physics: Definitions and Applications
### Section Key Terms
### What Physics Is
Think about all of the technological devices that you use on a regular basis. Computers, wireless internet, smart phones, tablets, global positioning system (GPS), MP3 players, and satellite radio might come to mind. Next, think about the most exciting modern technologies that you have heard about in the news, such as trains that levitate above their tracks, invisibility cloaks that bend light around them, and microscopic robots that fight diseased cells in our bodies. All of these groundbreaking advancements rely on the principles of physics.
Physics is a branch of science. The word science comes from a Latin word that means having knowledge, and refers to the knowledge of how the physical world operates, based on objective evidence determined through observation and experimentation. A key requirement of any scientific explanation of a natural phenomenon is that it must be testable; one must be able to devise and conduct an experimental investigation that either supports or refutes the explanation. It is important to note that some questions fall outside the realm of science precisely because they deal with phenomena that are not scientifically testable. This need for objective evidence helps define the investigative process scientists follow, which will be described later in this chapter.
Physics is the science aimed at describing the fundamental aspects of our universe. This includes what things are in it, what properties of those things are noticeable, and what processes those things or their properties undergo. In simpler terms, physics attempts to describe the basic mechanisms that make our universe behave the way it does. For example, consider a smart phone (). Physics describes how electric current interacts with the various circuits inside the device. This knowledge helps engineers select the appropriate materials and circuit layout when building the smart phone. Next, consider a GPS. Physics describes the relationship between the speed of an object, the distance over which it travels, and the time it takes to travel that distance. When you use a GPS device in a vehicle, it utilizes these physics relationships to determine the travel time from one location to another.
As our technology evolved over the centuries, physics expanded into many branches. Ancient peoples could only study things that they could see with the naked eye or otherwise experience without the aid of scientific equipment. This included the study of kinematics, which is the study of moving objects. For example, ancient people often studied the apparent motion of objects in the sky, such as the sun, moon, and stars. This is evident in the construction of prehistoric astronomical observatories, such as Stonehenge in England (shown in ).
Ancient people also studied statics and dynamics, which focus on how objects start moving, stop moving, and change speed and direction in response to forces that push or pull on the objects. This early interest in kinematics and dynamics allowed humans to invent simple machines, such as the lever, the pulley, the ramp, and the wheel. These simple machines were gradually combined and integrated to produce more complicated machines, such as wagons and cranes. Machines allowed humans to gradually do more work more effectively in less time, allowing them to create larger and more complicated buildings and structures, many of which still exist today from ancient times.
As technology advanced, the branches of physics diversified even more. These include branches such as acoustics, the study of sound, and optics, the study of the light. In 1608, the invention of the telescope by a Germany spectacle maker, Hans Lippershey, led to huge breakthroughs in astronomythe study of objects or phenomena in space. One year later, in 1609, Galileo Galilei began the first studies of the solar system and the universe using a telescope. During the Renaissance era, Isaac Newton used observations made by Galileo to construct his three laws of motion. These laws were the standard for studying kinematics and dynamics even today.
Another major branch of physics is thermodynamics, which includes the study of thermal energy and the transfer of heat. James Prescott Joule, an English physicist, studied the nature of heat and its relationship to work. Joule’s work helped lay the foundation for the first of three laws of thermodynamics that describe how energy in our universe is transferred from one object to another or transformed from one form to another. Studies in thermodynamics were motivated by the need to make engines more efficient, keep people safe from the elements, and preserve food.
The 18th and 19th centuries also saw great strides in the study of electricity and magnetism. Electricity involves the study of electric charges and their movements. Magnetism had long ago been noticed as an attractive force between a magnetized object and a metal like iron, or between the opposite poles (North and South) of two magnetized objects. In 1820, Danish physicist Hans Christian Oersted showed that electric currents create magnetic fields. In 1831, English inventor Michael Faraday showed that moving a wire through a magnetic field could induce an electric current. These studies led to the inventions of the electric motor and electric generator, which revolutionized human life by bringing electricity and magnetism into our machines.
The end of the 19th century saw the discovery of radioactive substances by the scientists Marie and Pierre Curie. Nuclear physics involves studying the nuclei of atoms, the source of nuclear radiation. In the 20th century, the study of nuclear physics eventually led to the ability to split the nucleus of an atom, a process called nuclear fission. This process is the basis for nuclear power plants and nuclear weapons. Also, the field of quantum mechanics, which involves the mechanics of atoms and molecules, saw great strides during the 20th century as our understanding of atoms and subatomic particles increased (see below).
Early in the 20th century, Albert Einstein revolutionized several branches of physics, especially relativity. Relativity revolutionized our understanding of motion and the universe in general as described further in this chapter. Now, in the 21st century, physicists continue to study these and many other branches of physics.
By studying the most important topics in physics, you will gain analytical abilities that will enable you to apply physics far beyond the scope of what can be included in a single book. These analytical skills will help you to excel academically, and they will also help you to think critically in any career you choose to pursue.
### Physics: Past and Present
The word physics is thought to come from the Greek word phusis, meaning nature. The study of nature later came to be called natural philosophy. From ancient times through the Renaissance, natural philosophy encompassed many fields, including astronomy, biology, chemistry, mathematics, and medicine. Over the last few centuries, the growth of scientific knowledge has resulted in ever-increasing specialization and branching of natural philosophy into separate fields, with physics retaining the most basic facets. Physics, as it developed from the Renaissance to the end of the 19th century, is called classical physics. Revolutionary discoveries starting at the beginning of the 20th century transformed physics from classical physics to modern physics.
Classical physics is not an exact description of the universe, but it is an excellent approximation under the following conditions: (1) matter must be moving at speeds less than about 1 percent of the speed of light, (2) the objects dealt with must be large enough to be seen with the naked eye, and (3) only weak gravity, such as that generated by Earth, can be involved. Very small objects, such as atoms and molecules, cannot be adequately explained by classical physics. These three conditions apply to almost all of everyday experience. As a result, most aspects of classical physics should make sense on an intuitive level.
Many laws of classical physics have been modified during the 20th century, resulting in revolutionary changes in technology, society, and our view of the universe. As a result, many aspects of modern physics, which occur outside of the range of our everyday experience, may seem bizarre or unbelievable. So why is most of this textbook devoted to classical physics? There are two main reasons. The first is that knowledge of classical physics is necessary to understand modern physics. The second reason is that classical physics still gives an accurate description of the universe under a wide range of everyday circumstances.
Modern physics includes two revolutionary theories: relativity and quantum mechanics. These theories deal with the very fast and the very small, respectively. The theory of relativity was developed by Albert Einstein in 1905. By examining how two observers moving relative to each other would see the same phenomena, Einstein devised radical new ideas about time and space. He came to the startling conclusion that the measured length of an object travelling at high speeds (greater than about one percent of the speed of light) is shorter than the same object measured at rest. Perhaps even more bizarre is the idea that the time for the same process to occur is different depending on the motion of the observer. Time passes more slowly for an object travelling at high speeds. A trip to the nearest star system, Alpha Centauri, might take an astronaut 4.5 Earth years if the ship travels near the speed of light. However, because time is slowed at higher speeds, the astronaut would age only 0.5 years during the trip. Einstein’s ideas of relativity were accepted after they were confirmed by numerous experiments.
Gravity, the force that holds us to Earth, can also affect time and space. For example, time passes more slowly on Earth’s surface than for objects farther from the surface, such as a satellite in orbit. The very accurate clocks on global positioning satellites have to correct for this. They slowly keep getting ahead of clocks at Earth’s surface. This is called time dilation, and it occurs because gravity, in essence, slows down time.
Large objects, like Earth, have strong enough gravity to distort space. To visualize this idea, think about a bowling ball placed on a trampoline. The bowling ball depresses or curves the surface of the trampoline. If you rolled a marble across the trampoline, it would follow the surface of the trampoline, roll into the depression caused by the bowling ball, and hit the ball. Similarly, the Earth curves space around it in the shape of a funnel. These curves in space due to the Earth cause objects to be attracted to Earth (i.e., gravity).
Because of the way gravity affects space and time, Einstein stated that gravity affects the space-time continuum, as illustrated in . This is why time proceeds more slowly at Earth’s surface than in orbit. In black holes, whose gravity is hundreds of times that of Earth, time passes so slowly that it would appear to a far-away observer to have stopped!
In summary, relativity says that in describing the universe, it is important to realize that time, space and speed are not absolute. Instead, they can appear different to different observers. Einstein’s ability to reason out relativity is even more amazing because we cannot see the effects of relativity in our everyday lives.
Quantum mechanics is the second major theory of modern physics. Quantum mechanics deals with the very small, namely, the subatomic particles that make up atoms. Atoms () are the smallest units of elements. However, atoms themselves are constructed of even smaller subatomic particles, such as protons, neutrons and electrons. Quantum mechanics strives to describe the properties and behavior of these and other subatomic particles. Often, these particles do not behave in the ways expected by classical physics.
At particle colliders (), such as the Large Hadron Collider on the France-Swiss border, particle physicists can make subatomic particles travel at very high speeds within a 27 kilometers (17 miles) long superconducting tunnel. They can then study the properties of the particles at high speeds, as well as collide them with each other to see how they exchange energy. This has led to many intriguing discoveries such as the Higgs-Boson particle, which gives matter the property of mass, and antimatter, which causes a huge energy release when it comes in contact with matter.
Physicists are currently trying to unify the two theories of modern physics, relativity and quantum mechanics, into a single, comprehensive theory called relativistic quantum mechanics. Relating the behavior of subatomic particles to gravity, time, and space will allow us to explain how the universe works in a much more comprehensive way.
### Application of Physics
You need not be a scientist to use physics. On the contrary, knowledge of physics is useful in everyday situations as well as in nonscientific professions. For example, physics can help you understand why you shouldn’t put metal in the microwave (), why a black car radiator helps remove heat in a car engine, and why a white roof helps keep the inside of a house cool. The operation of a car’s ignition system, as well as the transmission of electrical signals through our nervous system, are much easier to understand when you think about them in terms of the basic physics of electricity.
Physics is the foundation of many important scientific disciplines. For example, chemistry deals with the interactions of atoms and molecules. Not surprisingly, chemistry is rooted in atomic and molecular physics. Most branches of engineering are also applied physics. In architecture, physics is at the heart of determining structural stability, acoustics, heating, lighting, and cooling for buildings. Parts of geology, the study of nonliving parts of Earth, rely heavily on physics; including radioactive dating, earthquake analysis, and heat transfer across Earth’s surface. Indeed, some disciplines, such as biophysics and geophysics, are hybrids of physics and other disciplines.
Physics also describes the chemical processes that power the human body. Physics is involved in medical diagnostics, such as x-rays, magnetic resonance imaging (MRI), and ultrasonic blood flow measurements (). Medical therapy Physics also has many applications in biology, the study of life. For example, physics describes how cells can protect themselves using their cell walls and cell membranes (). Medical therapy sometimes directly involves physics, such as in using X-rays to diagnose health conditions. Physics can also explain what we perceive with our senses, such as how the ears detect sound or the eye detects color.
In summary, physics studies many of the most basic aspects of science. A knowledge of physics is, therefore, necessary to understand all other sciences. This is because physics explains the most basic ways in which our universe works. However, it is not necessary to formally study all applications of physics. A knowledge of the basic laws of physics will be most useful to you, so that you can use them to solve some everyday problems. In this way, the study of physics can improve your problem-solving skills.
### Check Your Understanding
### Section Summary
1. Physics is the most fundamental of the sciences, concerning itself with energy, matter, space and time, and their interactions.
2. Modern physics involves the theory of relativity, which describes how time, space and gravity are not constant in our universe can be different for different observers, and quantum mechanics, which describes the behavior of subatomic particles.
3. Physics is the basis for all other sciences, such as chemistry, biology and geology, because physics describes the fundamental way in which the universe functions.
### Concept Items
### Critical Thinking
### Test Prep Multiple Choice
### Test Prep Short Answer
|
# What is Physics?
## The Scientific Methods
### Section Key Terms
### Scientific Methods
Scientists often plan and carry out investigations to answer questions about the universe around us. These investigations may lead to natural laws. Such laws are intrinsic to the universe, meaning that humans did not create them and cannot change them. We can only discover and understand them. Their discovery is a very human endeavor, with all the elements of mystery, imagination, struggle, triumph, and disappointment inherent in any creative effort. The cornerstone of discovering natural laws is observation. Science must describe the universe as it is, not as we imagine or wish it to be.
We all are curious to some extent. We look around, make generalizations, and try to understand what we see. For example, we look up and wonder whether one type of cloud signals an oncoming storm. As we become serious about exploring nature, we become more organized and formal in collecting and analyzing data. We attempt greater precision, perform controlled experiments (if we can), and write down ideas about how data may be organized. We then formulate models, theories, and laws based on the data we have collected, and communicate those results with others. This, in a nutshell, describes the scientific method that scientists employ to decide scientific issues on the basis of evidence from observation and experiment.
An investigation often begins with a scientist making an observation. The scientist observes a pattern or trend within the natural world. Observation may generate questions that the scientist wishes to answer. Next, the scientist may perform some research about the topic and devise a hypothesis. A hypothesis is a testable statement that describes how something in the natural world works. In essence, a hypothesis is an educated guess that explains something about an observation.
Scientists may test the hypothesis by performing an experiment. During an experiment, the scientist collects data that will help them learn about the phenomenon they are studying. Then the scientists analyze the results of the experiment (that is, the data), often using statistical, mathematical, and/or graphical methods. From the data analysis, they draw conclusions. They may conclude that their experiment either supports or rejects their hypothesis. If the hypothesis is supported, the scientist usually goes on to test another hypothesis related to the first. If their hypothesis is rejected, they will often then test a new and different hypothesis in their effort to learn more about whatever they are studying.
Scientific processes can be applied to many situations. Let’s say that you try to turn on your car, but it will not start. You have just made an observation! You ask yourself, "Why won’t my car start?" You can now use scientific processes to answer this question. First, you generate a hypothesis such as, "The car won’t start because it has no gasoline in the gas tank." To test this hypothesis, you put gasoline in the car and try to start it again. If the car starts, then your hypothesis is supported by the experiment. If the car does not start, then your hypothesis is rejected. You will then need to think up a new hypothesis to test such as, "My car won’t start because the fuel pump is broken." Hopefully, your investigations lead you to discover why the car wont start and enable you to fix it.
### Modeling
A model is a representation of something that is often too difficult (or impossible) to study directly. Models can take the form of physical models, equations, computer programs, or simulations—computer graphics/animations. Models are tools that are especially useful in modern physics because they let us visualize phenomena that we normally cannot observe with our senses, such as very small objects or objects that move at high speeds. For example, we can understand the structure of an atom using models, without seeing an atom with our own eyes. Although images of single atoms are now possible, these images are extremely difficult to achieve and are only possible due to the success of our models. The existence of these images is a consequence rather than a source of our understanding of atoms. Models are always approximate, so they are simpler to consider than the real situation; the more complete a model is, the more complicated it must be. Models put the intangible or the extremely complex into human terms that we can visualize, discuss, and hypothesize about.
Scientific models are constructed based on the results of previous experiments. Even still, models often only describe a phenomenon partially or in a few limited situations. Some phenomena are so complex that they may be impossible to model them in their entirety, even using computers. An example is the electron cloud model of the atom in which electrons are moving around the atom’s center in distinct clouds (), that represent the likelihood of finding an electron in different places. This model helps us to visualize the structure of an atom. However, it does not show us exactly where an electron will be within its cloud at any one particular time.
As mentioned previously, physicists use a variety of models including equations, physical models, computer simulations, etc. For example, three-dimensional models are often commonly used in chemistry and physics to model molecules. Properties other than appearance or location are usually modelled using mathematics, where functions are used to show how these properties relate to one another. Processes such as the formation of a star or the planets, can also be modelled using computer simulations. Once a simulation is correctly programmed based on actual experimental data, the simulation can allow us to view processes that happened in the past or happen too quickly or slowly for us to observe directly. In addition, scientists can also run virtual experiments using computer-based models. In a model of planet formation, for example, the scientist could alter the amount or type of rocks present in space and see how it affects planet formation.
Scientists use models and experimental results to construct explanations of observations or design solutions to problems. For example, one way to make a car more fuel efficient is to reduce the friction or drag caused by air flowing around the moving car. This can be done by designing the body shape of the car to be more aerodynamic, such as by using rounded corners instead of sharp ones. Engineers can then construct physical models of the car body, place them in a wind tunnel, and examine the flow of air around the model. This can also be done mathematically in a computer simulation. The air flow pattern can be analyzed for regions smooth air flow and for eddies that indicate drag. The model of the car body may have to be altered slightly to produce the smoothest pattern of air flow (i.e., the least drag). The pattern with the least drag may be the solution to increasing fuel efficiency of the car. This solution might then be incorporated into the car design.
### Scientific Laws and Theories
A scientific law is a description of a pattern in nature that is true in all circumstances that have been studied. That is, physical laws are meant to be universal, meaning that they apply throughout the known universe. Laws are often also concise, whereas theories are more complicated. A law can be expressed in the form of a single sentence or mathematical equation. For example, Newton’s second law of motion, which relates the motion of an object to the force applied (F), the mass of the object (m), and the object’s acceleration (a), is simply stated using the equation
Scientific ideas and explanations that are true in many, but not all situations in the universe are usually called principles. An example is Pascal’s principle, which explains properties of liquids, but not solids or gases. However, the distinction between laws and principles is sometimes not carefully made in science.
A theory is an explanation for patterns in nature that is supported by much scientific evidence and verified multiple times by multiple researchers. While many people confuse theories with educated guesses or hypotheses, theories have withstood more rigorous testing and verification than hypotheses.
As a closing idea about scientific processes, we want to point out that scientific laws and theories, even those that have been supported by experiments for centuries, can still be changed by new discoveries. This is especially true when new technologies emerge that allow us to observe things that were formerly unobservable. Imagine how viewing previously invisible objects with a microscope or viewing Earth for the first time from space may have instantly changed our scientific theories and laws! What discoveries still await us in the future? The constant retesting and perfecting of our scientific laws and theories allows our knowledge of nature to progress. For this reason, many scientists are reluctant to say that their studies prove anything. By saying support instead of prove, it keeps the door open for future discoveries, even if they won’t occur for centuries or even millennia.
### Check Your Understanding
### Section Summary
1. Science seeks to discover and describe the underlying order and simplicity in nature.
2. The processes of science include observation, hypothesis, experiment, and conclusion.
3. Theories are scientific explanations that are supported by a large body experimental results.
4. Scientific laws are concise descriptions of the universe that are universally true.
### Concept Items
### Critical Thinking
### Test Prep Multiple Choice
### Test Prep Short Answer
### Test Prep Extended Response
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# What is Physics?
## The Language of Physics: Physical Quantities and Units
### Section Key Terms
### The Role of Units
Physicists, like other scientists, make observations and ask basic questions. For example, how big is an object? How much mass does it have? How far did it travel? To answer these questions, they make measurements with various instruments (e.g., meter stick, balance, stopwatch, etc.).
The measurements of physical quantities are expressed in terms of units, which are standardized values. For example, the length of a race, which is a physical quantity, can be expressed in meters (for sprinters) or kilometers (for long distance runners). Without standardized units, it would be extremely difficult for scientists to express and compare measured values in a meaningful way ().
All physical quantities in the International System of Units (SI) are expressed in terms of combinations of seven fundamental physical units, which are units for: length, mass, time, electric current, temperature, amount of a substance, and luminous intensity.
### SI Units: Fundamental and Derived Units
In any system of units, the units for some physical quantities must be defined through a measurement process. These are called the base quantities for that system and their units are the system’s base units. All other physical quantities can then be expressed as algebraic combinations of the base quantities. Each of these physical quantities is then known as a derived quantity and each unit is called a derived unit. The choice of base quantities is somewhat arbitrary, as long as they are independent of each other and all other quantities can be derived from them. Typically, the goal is to choose physical quantities that can be measured accurately to a high precision as the base quantities. The reason for this is simple. Since the derived units can be expressed as algebraic combinations of the base units, they can only be as accurate and precise as the base units from which they are derived.
Based on such considerations, the International Standards Organization recommends using seven base quantities, which form the International System of Quantities (ISQ). These are the base quantities used to define the SI base units. () lists these seven ISQ base quantities and the corresponding SI base units.
### The Meter
The SI unit for length is the meter (m). The definition of the meter has changed over time to become more accurate and precise. The meter was first defined in 1791 as 1/10,000,000 of the distance from the equator to the North Pole. This measurement was improved in 1889 by redefining the meter to be the distance between two engraved lines on a platinum-iridium bar. (The bar is now housed at the International Bureau of Weights and Measures, near Paris). By 1960, some distances could be measured more precisely by comparing them to wavelengths of light. The meter was redefined as 1,650,763.73 wavelengths of orange light emitted by krypton atoms. In 1983, the meter was given its present definition as the distance light travels in a vacuum in 1/ 299,792,458 of a second ().
### The Kilogram
The SI unit for mass is the kilogram (abbreviated kg); it was previously defined to be the mass of a platinum-iridium cylinder kept with the old meter standard at the International Bureau of Weights and Measures near Paris. Exact replicas of the previously defined kilogram are also kept at the United States’ National Institute of Standards and Technology, or NIST, located in Gaithersburg, Maryland outside of Washington D.C., and at other locations around the world. The determination of all other masses could be ultimately traced to a comparison with the standard mass. Even though the platinum-iridium cylinder was resistant to corrosion, airborne contaminants were able to adhere to its surface, slightly changing its mass over time. In May 2019, the scientific community adopted a more stable definition of the kilogram. The kilogram is now defined in terms of the second, the meter, and Planck's constant, h (a quantum mechanical value that relates a photon's energy to its frequency).
### The Second
The SI unit for time, the second (s) also has a long history. For many years it was defined as 1/86,400 of an average solar day. However, the average solar day is actually very gradually getting longer due to gradual slowing of Earth’s rotation. Accuracy in the fundamental units is essential, since all other measurements are derived from them. Therefore, a new standard was adopted to define the second in terms of a non-varying, or constant, physical phenomenon. One constant phenomenon is the very steady vibration of Cesium atoms, which can be observed and counted. This vibration forms the basis of the cesium atomic clock. In 1967, the second was redefined as the time required for 9,192,631,770 Cesium atom vibrations ().
### The Ampere
Electric current is measured in the ampere (A), named after Andre Ampere. You have probably heard of amperes, or amps, when people discuss electrical currents or electrical devices. Understanding an ampere requires a basic understanding of electricity and magnetism, something that will be explored in depth in later chapters of this book. Basically, two parallel wires with an electric current running through them will produce an attractive force on each other. One ampere is defined as the amount of electric current that will produce an attractive force of 2.7 10–7 newton per meter of separation between the two wires (the newton is the derived unit of force).
### Kelvins
The SI unit of temperature is the kelvin (or kelvins, but not degrees kelvin). This scale is named after physicist William Thomson, Lord Kelvin, who was the first to call for an absolute temperature scale. The Kelvin scale is based on absolute zero. This is the point at which all thermal energy has been removed from all atoms or molecules in a system. This temperature, 0 K, is equal to −273.15 °C and −459.67 °F. Conveniently, the Kelvin scale actually changes in the same way as the Celsius scale. For example, the freezing point (0 °C) and boiling points of water (100 °C) are 100 degrees apart on the Celsius scale. These two temperatures are also 100 kelvins apart (freezing point = 273.15 K; boiling point = 373.15 K).
### Metric Prefixes
Physical objects or phenomena may vary widely. For example, the size of objects varies from something very small (like an atom) to something very large (like a star). Yet the standard metric unit of length is the meter. So, the metric system includes many prefixes that can be attached to a unit. Each prefix is based on factors of 10 (10, 100, 1,000, etc., as well as 0.1, 0.01, 0.001, etc.). gives the metric prefixes and symbols used to denote the different various factors of 10 in the metric system.
The metric system is convenient because conversions between metric units can be done simply by moving the decimal place of a number. This is because the metric prefixes are sequential powers of 10. There are 100 centimeters in a meter, 1000 meters in a kilometer, and so on. In nonmetric systems, such as U.S. customary units, the relationships are less simple—there are 12 inches in a foot, 5,280 feet in a mile, 4 quarts in a gallon, and so on. Another advantage of the metric system is that the same unit can be used over extremely large ranges of values simply by switching to the most-appropriate metric prefix. For example, distances in meters are suitable for building construction, but kilometers are used to describe road construction. Therefore, with the metric system, there is no need to invent new units when measuring very small or very large objects—you just have to move the decimal point (and use the appropriate prefix).
### Known Ranges of Length, Mass, and Time
lists known lengths, masses, and time measurements. You can see that scientists use a range of measurement units. This wide range demonstrates the vastness and complexity of the universe, as well as the breadth of phenomena physicists study. As you examine this table, note how the metric system allows us to discuss and compare an enormous range of phenomena, using one system of measurement ( and ).
### Using Scientific Notation with Physical Measurements
Scientific notation is a way of writing numbers that are too large or small to be conveniently written as a decimal. For example, consider the number 840,000,000,000,000. It’s a rather large number to write out. The scientific notation for this number is 8.40 1014. Scientific notation follows this general format
In this format x is the value of the measurement with all placeholder zeros removed. In the example above, x is 8.4. The x is multiplied by a factor, 10, which indicates the number of placeholder zeros in the measurement. Placeholder zeros are those at the end of a number that is 10 or greater, and at the beginning of a decimal number that is less than 1. In the example above, the factor is 1014. This tells you that you should move the decimal point 14 positions to the right, filling in placeholder zeros as you go. In this case, moving the decimal point 14 places creates only 13 placeholder zeros, indicating that the actual measurement value is 840,000,000,000,000.
Numbers that are fractions can be indicated by scientific notation as well. Consider the number 0.0000045. Its scientific notation is 4.5 10–6. Its scientific notation has the same format
Here, x is 4.5. However, the value of y in the 10 factor is negative, which indicates that the measurement is a fraction of 1. Therefore, we move the decimal place to the left, for a negative y. In our example of 4.5 10–6, the decimal point would be moved to the left six times to yield the original number, which would be 0.0000045.
The term order of magnitude refers to the power of 10 when numbers are expressed in scientific notation. Quantities that have the same power of 10 when expressed in scientific notation, or come close to it, are said to be of the same order of magnitude. For example, the number 800 can be written as 8 102, and the number 450 can be written as 4.5 102. Both numbers have the same value for y. Therefore, 800 and 450 are of the same order of magnitude. Similarly, 101 and 99 would be regarded as the same order of magnitude, 102. Order of magnitude can be thought of as a ballpark estimate for the scale of a value. The diameter of an atom is on the order of 10−9 m, while the diameter of the sun is on the order of 109 m. These two values are 18 orders of magnitude apart.
Scientists make frequent use of scientific notation because of the vast range of physical measurements possible in the universe, such as the distance from Earth to the moon (), or to the nearest star.
### Unit Conversion and Dimensional Analysis
It is often necessary to convert from one type of unit to another. For example, if you are reading a European cookbook in the United States, some quantities may be expressed in liters and you need to convert them to cups. A Canadian tourist driving through the United States might want to convert miles to kilometers, to have a sense of how far away his next destination is. A doctor in the United States might convert a patient’s weight in pounds to kilograms.
Let’s consider a simple example of how to convert units within the metric system. How can we convert 1 hour to seconds?
First, we need to determine a conversion factor. A conversion factor is a ratio expressing how many of one unit are equal to another unit. A conversion factor is simply a fraction which equals 1. You can multiply any number by 1 and get the same value. When you multiply a number by a conversion factor, you are simply multiplying it by one. For example, the following are conversion factors: (1 foot)/(12 inches) = 1 to convert inches to feet, (1 meter)/(100 centimeters) = 1 to convert centimeters to meters, (1 minute)/(60 seconds) = 1 to convert seconds to minutes.
Now we can set up our unit conversion. We will write the units that we have and then multiply them by the conversion factor (1 km/1,000m) = 1, so we are simply multiplying 80m by 1:
When there is a unit in the original number, and a unit in the denominator (bottom) of the conversion factor, the units cancel. In this case, hours and minutes cancel and the value in seconds remains.
You can use this method to convert between any types of unit, including between the U.S. customary system and metric system. Notice also that, although you can multiply and divide units algebraically, you cannot add or subtract different units. An expression like 10 km + 5 kg makes no sense. Even adding two lengths in different units, such as 10 km + 20 m does not make sense. You express both lengths in the same unit. See Appendix C for a more complete list of conversion factors.
### Accuracy, Precision and Significant Figures
Science is based on experimentation that requires good measurements. The validity of a measurement can be described in terms of its accuracy and its precision (see and ). Accuracy is how close a measurement is to the correct value for that measurement. For example, let us say that you are measuring the length of standard piece of printer paper. The packaging in which you purchased the paper states that it is 11 inches long, and suppose this stated value is correct. You measure the length of the paper three times and obtain the following measurements: 11.1 inches, 11.2 inches, and 10.9 inches. These measurements are quite accurate because they are very close to the correct value of 11.0 inches. In contrast, if you had obtained a measurement of 12 inches, your measurement would not be very accurate. This is why measuring instruments are calibrated based on a known measurement. If the instrument consistently returns the correct value of the known measurement, it is safe for use in finding unknown values.
Precision states how well repeated measurements of something generate the same or similar results. Therefore, the precision of measurements refers to how close together the measurements are when you measure the same thing several times. One way to analyze the precision of measurements would be to determine the range, or difference between the lowest and the highest measured values. In the case of the printer paper measurements, the lowest value was 10.9 inches and the highest value was 11.2 inches. Thus, the measured values deviated from each other by, at most, 0.3 inches. These measurements were reasonably precise because they varied by only a fraction of an inch. However, if the measured values had been 10.9 inches, 11.1 inches, and 11.9 inches, then the measurements would not be very precise because there is a lot of variation from one measurement to another.
The measurements in the paper example are both accurate and precise, but in some cases, measurements are accurate but not precise, or they are precise but not accurate. Let us consider a GPS system that is attempting to locate the position of a restaurant in a city. Think of the restaurant location as existing at the center of a bull’s-eye target. Then think of each GPS attempt to locate the restaurant as a black dot on the bull’s eye.
In , you can see that the GPS measurements are spread far apart from each other, but they are all relatively close to the actual location of the restaurant at the center of the target. This indicates a low precision, high accuracy measuring system. However, in , the GPS measurements are concentrated quite closely to one another, but they are far away from the target location. This indicates a high precision, low accuracy measuring system. Finally, in , the GPS is both precise and accurate, allowing the restaurant to be located.
### Uncertainty
The accuracy and precision of a measuring system determine the uncertainty of its measurements. Uncertainty is a way to describe your confidence in your measured value, or the range of values that would be consistent with the data. If your measurements are not very accurate or precise, then the uncertainty of your values will be very high. In more general terms, uncertainty can be thought of as a disclaimer for your measured values. For example, if someone asked you to provide the mileage on your car, you might say that it is 45,000 miles, plus or minus 500 miles. The plus or minus amount is the uncertainty in your value. That is, you are indicating that the actual mileage of your car might be as low as 44,500 miles or as high as 45,500 miles, or anywhere in between. All measurements contain some amount of uncertainty. In our example of measuring the length of the paper, we might say that the length of the paper is 11 inches plus or minus 0.2 inches or 11.0 ± 0.2 inches. The uncertainty in a measurement, A, is often denoted as δA ("delta A"). The actual value of the object may not be within the range given by the measurement and its uncertainty. In our paper length example above, a new set of measurements might produce a length of 14.0 ± 0.2 inches, with the uncertainty based on the accuracy or our reading or repeated measurements. We would also, however, conclude that either one of our measurement sets is incorrect due to an offset in the measurement process in that set, or our measurement correctly identifies that we are measuring different papers. In the former case, the discrepancy between the measured value and the actual value is called a systematic error.
The factors contributing to uncertainty in a measurement include the following:
1. Limitations of the measuring device
2. The skill of the person making the measurement
3. Irregularities in the object being measured
4. Any other factors that affect the outcome (highly dependent on the situation)
In the printer paper example uncertainty could be caused by: the fact that the smallest division on the ruler is 0.1 inches, the person using the ruler has bad eyesight, or uncertainty caused by the paper cutting machine (e.g., one side of the paper is slightly longer than the other.) It is good practice to carefully consider all possible sources of uncertainty in a measurement and reduce or eliminate them.
### Percent Uncertainty
One method of expressing uncertainty is as a percent of the measured value. If a measurement, A, is expressed with uncertainty, δA, the percent uncertainty is
### Uncertainty in Calculations
There is an uncertainty in anything calculated from measured quantities. For example, the area of a floor calculated from measurements of its length and width has an uncertainty because the both the length and width have uncertainties. How big is the uncertainty in something you calculate by multiplication or division? If the measurements in the calculation have small uncertainties (a few percent or less), then the method of adding percents can be used. This method says that the percent uncertainty in a quantity calculated by multiplication or division is the sum of the percent uncertainties in the items used to make the calculation. For example, if a floor has a length of 4.00 m and a width of 3.00 m, with uncertainties of 2 percent and 1 percent, respectively, then the area of the floor is 12.0 m2 and has an uncertainty of 3 percent (expressed as an area this is 0.36 m2, which we round to 0.4 m2 since the area of the floor is given to a tenth of a square meter).
For more information on the accuracy, precision, and uncertainty of measurements based upon the units of measurement, visit this website.
### Precision of Measuring Tools and Significant Figures
An important factor in the accuracy and precision of measurements is the precision of the measuring tool. In general, a precise measuring tool is one that can measure values in very small increments. For example, consider measuring the thickness of a coin. A standard ruler can measure thickness to the nearest millimeter, while a micrometer can measure the thickness to the nearest 0.005 millimeter. The micrometer is a more precise measuring tool because it can measure extremely small differences in thickness. The more precise the measuring tool, the more precise and accurate the measurements can be.
When we express measured values, we can only list as many digits as we initially measured with our measuring tool (such as the rulers shown in ). For example, if you use a standard ruler to measure the length of a stick, you may measure it with a decimeter ruler as 3.6 cm. You could not express this value as 3.65 cm because your measuring tool was not precise enough to measure a hundredth of a centimeter. It should be noted that the last digit in a measured value has been estimated in some way by the person performing the measurement. For example, the person measuring the length of a stick with a ruler notices that the stick length seems to be somewhere in between 36 mm and 37 mm. He or she must estimate the value of the last digit. The rule is that the last digit written down in a measurement is the first digit with some uncertainty. For example, the last measured value 36.5 mm has three digits, or three significant figures. The number of significant figures in a measurement indicates the precision of the measuring tool. The more precise a measuring tool is, the greater the number of significant figures it can report.
### Zeros
Special consideration is given to zeros when counting significant figures. For example, the zeros in 0.053 are not significant because they are only placeholders that locate the decimal point. There are two significant figures in 0.053—the 5 and the 3. However, if the zero occurs between other significant figures, the zeros are significant. For example, both zeros in 10.053 are significant, as these zeros were actually measured. Therefore, the 10.053 placeholder has five significant figures. The zeros in 1300 may or may not be significant, depending on the style of writing numbers. They could mean the number is known to the last zero, or the zeros could be placeholders. So 1300 could have two, three, or four significant figures. To avoid this ambiguity, write 1300 in scientific notation as 1.3 × 103. Only significant figures are given in the x factor for a number in scientific notation (in the form ). Therefore, we know that 1 and 3 are the only significant digits in this number. In summary, zeros are significant except when they serve only as placeholders. provides examples of the number of significant figures in various numbers.
### Significant Figures in Calculations
When combining measurements with different degrees of accuracy and precision, the number of significant digits in the final answer can be no greater than the number of significant digits in the least precise measured value. There are two different rules, one for multiplication and division and another rule for addition and subtraction, as discussed below.
1.
But because the radius has only two significant figures, the area calculated is meaningful only to two significant figures or
even though the value of
2.
The least precise measurement is 13.7 kg. This measurement is expressed to the 0.1 decimal place, so our final answer must also be expressed to the 0.1 decimal place. Thus, the answer should be rounded to the tenths place, giving 15.2 kg. The same is true for non-decimal numbers. For example,
We cannot report the decimal places in the answer because
It is a good idea to keep extra significant figures while calculating, and to round off to the correct number of significant figures only in the final answers. The reason is that small errors from rounding while calculating can sometimes produce significant errors in the final answer. As an example, try calculating
### Significant Figures in this Text
In this textbook, most numbers are assumed to have three significant figures. Furthermore, consistent numbers of significant figures are used in all worked examples. You will note that an answer given to three digits is based on input good to at least three digits. If the input has fewer significant figures, the answer will also have fewer significant figures. Care is also taken that the number of significant figures is reasonable for the situation posed. In some topics, such as optics, more than three significant figures will be used. Finally, if a number is exact, such as the 2 in the formula,
, it does not affect the number of significant figures in a calculation.
In the example above, the final approximate value is much higher than the first friend’s early estimate of 3 in. However, the other friend’s early estimate of 10 ft. (120 in.) was roughly correct. How did the approximation measure up to your first guess? What can this exercise suggest about the value of rough guesstimates versus carefully calculated approximations?
### Graphing in Physics
Most results in science are presented in scientific journal articles using graphs. Graphs present data in a way that is easy to visualize for humans in general, especially someone unfamiliar with what is being studied. They are also useful for presenting large amounts of data or data with complicated trends in an easily-readable way.
One commonly-used graph in physics and other sciences is the line graph, probably because it is the best graph for showing how one quantity changes in response to the other. Let’s build a line graph based on the data in , which shows the measured distance that a train travels from its station versus time. Our two variables, or things that change along the graph, are time in minutes, and distance from the station, in kilometers. Remember that measured data may not have perfect accuracy.
1. Draw the two axes. The horizontal axis, or x-axis, shows the independent variable, which is the variable that is controlled or manipulated. The vertical axis, or y-axis, shows the dependent variable, the non-manipulated variable that changes with (or is dependent on) the value of the independent variable. In the data above, time is the independent variable and should be plotted on the x-axis. Distance from the station is the dependent variable and should be plotted on the y-axis.
2. Label each axes on the graph with the name of each variable, followed by the symbol for its units in parentheses. Be sure to leave room so that you can number each axis. In this example, use Time (min) as the label for the x-axis.
3.
Next, you must determine the best scale to use for numbering each axis. Because the time values on the
In general, you want to pick a scale for both axes that 1) shows all of your data, and 2) makes it easy to identify trends in your data. If you make your scale too large, it will be harder to see how your data change. Likewise, the smaller and more fine you make your scale, the more space you will need to make the graph. The number of significant figures in the axis values should be coarser than the number of significant figures in the measurements.
4. Now that your axes are ready, you can begin plotting your data. For the first data point, count along the x-axis until you find the 10 min tick mark. Then, count up from that point to the 10 km tick mark on the y-axis, and approximate where 22 km is along the y-axis. Place a dot at this location. Repeat for the other six data points ().
5. Add a title to the top of the graph to state what the graph is describing, such as the y-axis parameter vs. the x-axis parameter. In the graph shown here, the title is train motion. It could also be titled distance of the train from the station vs. time.
6. Finally, with data points now on the graph, you should draw a trend line (). The trend line represents the dependence you think the graph represents, so that the person who looks at your graph can see how close it is to the real data. In the present case, since the data points look like they ought to fall on a straight line, you would draw a straight line as the trend line. Draw it to come closest to all the points. Real data may have some inaccuracies, and the plotted points may not all fall on the trend line. In some cases, none of the data points fall exactly on the trend line.
### Analyzing a Graph Using Its Equation
One way to get a quick snapshot of a dataset is to look at the equation of its trend line. If the graph produces a straight line, the equation of the trend line takes the form
The b in the equation is the y-intercept while the m in the equation is the slope. The y-intercept tells you at what y value the line intersects the y-axis. In the case of the graph above, the y-intercept occurs at 0, at the very beginning of the graph. The y-intercept, therefore, lets you know immediately where on the y-axis the plot line begins.
The m in the equation is the slope. This value describes how much the line on the graph moves up or down on the y-axis along the line’s length. The slope is found using the following equation
In order to solve this equation, you need to pick two points on the line (preferably far apart on the line so the slope you calculate describes the line accurately). The quantities Y2 and Y1 represent the y-values from the two points on the line (not data points) that you picked, while X2 and X1 represent the two x-values of the those points.
What can the slope value tell you about the graph? The slope of a perfectly horizontal line will equal zero, while the slope of a perfectly vertical line will be undefined because you cannot divide by zero. A positive slope indicates that the line moves up the y-axis as the x-value increases while a negative slope means that the line moves down the y-axis. The more negative or positive the slope is, the steeper the line moves up or down, respectively. The slope of our graph in is calculated below based on the two endpoints of the line
Equation of line:
Because the x axis is time in minutes, we would actually be more likely to use the time t as the independent (x-axis) variable and write the equation as
The formula
only applies to linear relationships, or ones that produce a straight line. Another common type of line in physics is the quadratic relationship, which occurs when one of the variables is squared. One quadratic relationship in physics is the relation between the speed of an object its centripetal acceleration, which is used to determine the force needed to keep an object moving in a circle. Another common relationship in physics is the inverse relationship, in which one variable decreases whenever the other variable increases. An example in physics is Coulomb’s law. As the distance between two charged objects increases, the electrical force between the two charged objects decreases. Inverse proportionality, such the relation between x and y in the equation
for some number k, is one particular kind of inverse relationship. A third commonly-seen relationship is the exponential relationship, in which a change in the independent variable produces a proportional change in the dependent variable. As the value of the dependent variable gets larger, its rate of growth also increases. For example, bacteria often reproduce at an exponential rate when grown under ideal conditions. As each generation passes, there are more and more bacteria to reproduce. As a result, the growth rate of the bacterial population increases every generation ().
### Using Logarithmic Scales in Graphing
Sometimes a variable can have a very large range of values. This presents a problem when you’re trying to figure out the best scale to use for your graph’s axes. One option is to use a logarithmic (log) scale. In a logarithmic scale, the value each mark labels is the previous mark’s value multiplied by some constant. For a log base 10 scale, each mark labels a value that is 10 times the value of the mark before it. Therefore, a base 10 logarithmic scale would be numbered: 0, 10, 100, 1,000, etc. You can see how the logarithmic scale covers a much larger range of values than the corresponding linear scale, in which the marks would label the values 0, 10, 20, 30, and so on.
If you use a logarithmic scale on one axis of the graph and a linear scale on the other axis, you are using a semi-log plot. The Richter scale, which measures the strength of earthquakes, uses a semi-log plot. The degree of ground movement is plotted on a logarithmic scale against the assigned intensity level of the earthquake, which ranges linearly from 1-10 ( (a)).
If a graph has both axes in a logarithmic scale, then it is referred to as a log-log plot. The relationship between the wavelength and frequency of electromagnetic radiation such as light is usually shown as a log-log plot ( (b)). Log-log plots are also commonly used to describe exponential functions, such as radioactive decay.
### Check Your Understanding
### Section Summary
1. Physical quantities are a characteristic or property of an object that can be measured or calculated from other measurements.
2. The four fundamental units we will use in this textbook are the meter (for length), the kilogram (for mass), the second (for time), and the ampere (for electric current). These units are part of the metric system, which uses powers of 10 to relate quantities over the vast ranges encountered in nature.
3. Unit conversions involve changing a value expressed in one type of unit to another type of unit. This is done by using conversion factors, which are ratios relating equal quantities of different units.
4. Accuracy of a measured value refers to how close a measurement is to the correct value. The uncertainty in a measurement is an estimate of the amount by which the measurement result may differ from this value.
5. Precision of measured values refers to how close the agreement is between repeated measurements.
6. Significant figures express the precision of a measuring tool.
7. When multiplying or dividing measured values, the final answer can contain only as many significant figures as the least precise value.
8. When adding or subtracting measured values, the final answer cannot contain more decimal places than the least precise value.
### Key Equations
### Concept Items
### Problems
### Critical Thinking
### Performance Task
### Test Prep Multiple Choice
### Test Prep Short Answer
### Test Prep Extended Response
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# Motion in One Dimension
## Introduction
Outside of an airplane, have you ever traveled faster than 150 mph? Can you imagine traveling in a train like the one shown in that goes close to 300 mph? Despite the high speed, the people riding in this train may not notice that they are moving at all unless they look out the window! This is because motion, even motion at 300 mph, is relative to the observer.
In this chapter, you will learn why it is important to identify a reference frame in order to clearly describe motion. For now, the motion you describe will be one-dimensional. Within this context, you will learn the difference between distance and displacement as well as the difference between speed and velocity. Then you will look at some graphing and problem-solving techniques. |
# Motion in One Dimension
## Relative Motion, Distance, and Displacement
### Section Key Terms
### Defining Motion
Our study of physics opens with kinematics—the study of motion without considering its causes. Objects are in motion everywhere you look. Everything from a tennis game to a space-probe flyby of the planet Neptune involves motion. When you are resting, your heart moves blood through your veins. Even in inanimate objects, atoms are always moving.
How do you know something is moving? The location of an object at any particular time is its position. More precisely, you need to specify its position relative to a convenient reference frame. Earth is often used as a reference frame, and we often describe the position of an object as it relates to stationary objects in that reference frame. For example, a rocket launch would be described in terms of the position of the rocket with respect to Earth as a whole, while a professor’s position could be described in terms of where she is in relation to the nearby white board. In other cases, we use reference frames that are not stationary but are in motion relative to Earth. To describe the position of a person in an airplane, for example, we use the airplane, not Earth, as the reference frame. (See .) Thus, you can only know how fast and in what direction an object's position is changing against a background of something else that is either not moving or moving with a known speed and direction. The reference frame is the coordinate system from which the positions of objects are described.
Your classroom can be used as a reference frame. In the classroom, the walls are not moving. Your motion as you walk to the door, can be measured against the stationary background of the classroom walls. You can also tell if other things in the classroom are moving, such as your classmates entering the classroom or a book falling off a desk. You can also tell in what direction something is moving in the classroom. You might say, “The teacher is moving toward the door.” Your reference frame allows you to determine not only that something is moving but also the direction of motion.
You could also serve as a reference frame for others’ movement. If you remained seated as your classmates left the room, you would measure their movement away from your stationary location. If you and your classmates left the room together, then your perspective of their motion would be change. You, as the reference frame, would be moving in the same direction as your other moving classmates. As you will learn in the Snap Lab, your description of motion can be quite different when viewed from different reference frames.
### Distance vs. Displacement
As we study the motion of objects, we must first be able to describe the object’s position. Before your parent drives you to school, the car is sitting in your driveway. Your driveway is the starting position for the car. When you reach your high school, the car has changed position. Its new position is your school.
Physicists use variables to represent terms. We will use d to represent car’s position. We will use a subscript to differentiate between the initial position, d0, and the final position, df. In addition, vectors, which we will discuss later, will be in bold or will have an arrow above the variable. Scalars will be italicized.
Now imagine driving from your house to a friend's house located several kilometers away. How far would you drive? The distance an object moves is the length of the path between its initial position and its final position. The distance you drive to your friend's house depends on your path. As shown in , distance is different from the length of a straight line between two points. The distance you drive to your friend's house is probably longer than the straight line between the two houses.
We often want to be more precise when we talk about position. The description of an object’s motion often includes more than just the distance it moves. For instance, if it is a five kilometer drive to school, the distance traveled is 5 kilometers. After dropping you off at school and driving back home, your parent will have traveled a total distance of 10 kilometers. The car and your parent will end up in the same starting position in space. The net change in position of an object is its displacement, or
The Greek letter delta,
, means change in.
If you are describing only your drive to school and the route is a straight line, then the distance traveled and the displacement are the same—5 kilometers. When you are describing the entire round trip, distance and displacement are different. When you describe distance, you only include the magnitude, the size or amount, of the distance traveled. However, when you describe the displacement, you take into account both the magnitude of the change in position and the direction of movement.
In our previous example, the car travels a total of 10 kilometers, but it drives five of those kilometers forward toward school and five of those kilometers back in the opposite direction. If we ascribe the forward direction a positive (+) and the opposite direction a negative (–), then the two quantities will cancel each other out when added together.
A quantity, such as distance, that has magnitude (i.e., how big or how much) and sometimes a sign (e.g., electric charge, temperature in Celsius, or component of a vector) but does not take into account direction is called a scalar. A quantity, such as displacement, that has both magnitude and direction is called a vector. A vector with magnitude zero is a special case of a vector that has no direction.
### Displacement Problems
Hopefully you now understand the conceptual difference between distance and displacement. Understanding concepts is half the battle in physics. The other half is math. A stumbling block to new physics students is trying to wade through the math of physics while also trying to understand the associated concepts. This struggle may lead to misconceptions and answers that make no sense. Once the concept is mastered, the math is far less confusing.
So let’s review and see if we can make sense of displacement in terms of numbers and equations. You can calculate an object's displacement by subtracting its original position, d, from its final position d. In math terms that means
If the final position is the same as the initial position, then
.
To assign numbers and/or direction to these quantities, we need to define an axis with a positive and a negative direction. We also need to define an origin, or O. In , the axis is in a straight line with home at zero and school in the positive direction. If we left home and drove the opposite way from school, motion would have been in the negative direction. We would have assigned it a negative value. In the round-trip drive, df and d0 were both at zero kilometers. In the one way trip to school, df was at 5 kilometers and d0 was at zero km. So,
was 5 kilometers.
### Practice Problems
### Check Your Understanding
### Section Summary
1. A description of motion depends on the reference frame from which it is described.
2. The distance an object moves is the length of the path along which it moves.
3. Displacement is the difference in the initial and final positions of an object.
### Key Equations
### Concept Items
### Critical Thinking
### Problems
### Test Prep Multiple Choice
### Test Prep Short Answer
### Test Prep Extended Response
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# Motion in One Dimension
## Speed and Velocity
### Section Key Terms
### Speed
There is more to motion than distance and displacement. Questions such as, “How long does a foot race take?” and “What was the runner’s speed?” cannot be answered without an understanding of other concepts. In this section we will look at time, speed, and velocity to expand our understanding of motion.
A description of how fast or slow an object moves is its speed. Speed is the rate at which an object changes its location. Like distance, speed is a scalar because it has a magnitude but not a direction. Because speed is a rate, it depends on the time interval of motion. You can calculate the elapsed time or the change in time,
, of motion as the difference between the ending time and the beginning time
The SI unit of time is the second (s), and the SI unit of speed is meters per second (m/s), but sometimes kilometers per hour (km/h), miles per hour (mph) or other units of speed are used.
When you describe an object's speed, you often describe the average over a time period. Average speed, v, is the distance traveled divided by the time during which the motion occurs.
You can, of course, rearrange the equation to solve for either distance or time
Suppose, for example, a car travels 150 kilometers in 3.2 hours. Its average speed for the trip is
A car's speed would likely increase and decrease many times over a 3.2 hour trip. Its speed at a specific instant in time, however, is its instantaneous speed. A car's speedometer describes its instantaneous speed.
### Practice Problems
### Velocity
The vector version of speed is velocity. Velocity describes the speed and direction of an object. As with speed, it is useful to describe either the average velocity over a time period or the velocity at a specific moment. Average velocity is displacement divided by the time over which the displacement occurs.
Velocity, like speed, has SI units of meters per second (m/s), but because it is a vector, you must also include a direction. Furthermore, the variable v for velocity is bold because it is a vector, which is in contrast to the variable v for speed which is italicized because it is a scalar quantity.
Suppose a passenger moved toward the back of a plane with an average velocity of –4 m/s. We cannot tell from the average velocity whether the passenger stopped momentarily or backed up before he got to the back of the plane. To get more details, we must consider smaller segments of the trip over smaller time intervals such as those shown in . If you consider infinitesimally small intervals, you can define instantaneous velocity, which is the velocity at a specific instant in time. Instantaneous velocity and average velocity are the same if the velocity is constant.
Earlier, you have read that distance traveled can be different than the magnitude of displacement. In the same way, speed can be different than the magnitude of velocity. For example, you drive to a store and return home in half an hour. If your car’s odometer shows the total distance traveled was 6 km, then your average speed was 12 km/h. Your average velocity, however, was zero because your displacement for the round trip is zero.
### Practice Problems
### Check Your Understanding
### Section Summary
1. Average speed is a scalar quantity that describes distance traveled divided by the time during which the motion occurs.
2. Velocity is a vector quantity that describes the speed and direction of an object.
3. Average velocity is displacement over the time period during which the displacement occurs. If the velocity is constant, then average velocity and instantaneous velocity are the same.
### Key Equations
### Concept Items
### Problems
### Critical Thinking
### Test Prep Multiple Choice
### Test Prep Short Answer
### Test Prep Extended Response
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# Motion in One Dimension
## Position vs. Time Graphs
### Section Key Terms
### Graphing Position as a Function of Time
A graph, like a picture, is worth a thousand words. Graphs not only contain numerical information, they also reveal relationships between physical quantities. In this section, we will investigate kinematics by analyzing graphs of position over time.
Graphs in this text have perpendicular axes, one horizontal and the other vertical. When two physical quantities are plotted against each other, the horizontal axis is usually considered the independent variable, and the vertical axis is the dependent variable. In algebra, you would have referred to the horizontal axis as the x-axis and the vertical axis as the y-axis. As in , a straight-line graph has the general form
.
Here m is the slope, defined as the rise divided by the run (as seen in the figure) of the straight line. The letter b is the y-intercept which is the point at which the line crosses the vertical, y-axis. In terms of a physical situation in the real world, these quantities will take on a specific significance, as we will see below. (.)
In physics, time is usually the independent variable. Other quantities, such as displacement, are said to depend upon it. A graph of position versus time, therefore, would have position on the vertical axis (dependent variable) and time on the horizontal axis (independent variable). In this case, to what would the slope and y-intercept refer? Let’s look back at our original example when studying distance and displacement.
The drive to school was 5 km from home. Let’s assume it took 10 minutes to make the drive and that your parent was driving at a constant velocity the whole time. The position versus time graph for this section of the trip would look like that shown in .
As we said before, d0 = 0 because we call home our O and start calculating from there. In , the line starts at d = 0, as well. This is the b in our equation for a straight line. Our initial position in a position versus time graph is always the place where the graph crosses the x-axis at t = 0. What is the slope? The rise is the change in position, (i.e., displacement) and the run is the change in time. This relationship can also be written
This relationship was how we defined average velocity. Therefore, the slope in a d versus t graph, is the average velocity.
Let’s look at another example. shows a graph of position versus time for a jet-powered car on a very flat dry lake bed in Nevada.
Using the relationship between dependent and independent variables, we see that the slope in the graph in is average velocity, vavg and the intercept is displacement at time zero—that is, d0. Substituting these symbols into y = mx + b gives
or
Thus a graph of position versus time gives a general relationship among displacement, velocity, and time, as well as giving detailed numerical information about a specific situation. From the figure we can see that the car has a position of 400 m at t = 0 s, 650 m at t = 1.0 s, and so on. And we can learn about the object’s velocity, as well.
### Solving Problems Using Position vs. Time Graphs
So how do we use graphs to solve for things we want to know like velocity?
But what if the graph of the position is more complicated than a straight line? What if the object speeds up or turns around and goes backward? Can we figure out anything about its velocity from a graph of that kind of motion? Let’s take another look at the jet-powered car. The graph in shows its motion as it is getting up to speed after starting at rest. Time starts at zero for this motion (as if measured with a stopwatch), and the displacement and velocity are initially 200 m and 15 m/s, respectively.
The graph of position versus time in is a curve rather than a straight line. The slope of the curve becomes steeper as time progresses, showing that the velocity is increasing over time. The slope at any point on a position-versus-time graph is the instantaneous velocity at that point. It is found by drawing a straight line tangent to the curve at the point of interest and taking the slope of this straight line. Tangent lines are shown for two points in . The average velocity is the net displacement divided by the time traveled.
### Practice Problems
### Check Your Understanding
### Section Summary
1. Graphs can be used to analyze motion.
2. The slope of a position vs. time graph is the velocity.
3. For a straight line graph of position, the slope is the average velocity.
4. To obtain the instantaneous velocity at a given moment for a curved graph, find the tangent line at that point and take its slope.
### Key Equations
### Concept Items
### Critical Thinking
### Problems
### Test Prep Multiple Choice
### Test Prep Short Answer
### Test Prep Extended Response
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# Motion in One Dimension
## Velocity vs. Time Graphs
### Section Key Terms
### Graphing Velocity as a Function of Time
Earlier, we examined graphs of position versus time. Now, we are going to build on that information as we look at graphs of velocity vs. time. Velocity is the rate of change of displacement. Acceleration is the rate of change of velocity; we will discuss acceleration more in another chapter. These concepts are all very interrelated.
What can we learn about motion by looking at velocity vs. time graphs? Let’s return to our drive to school, and look at a graph of position versus time as shown in .
We assumed for our original calculation that your parent drove with a constant velocity to and from school. We now know that the car could not have gone from rest to a constant velocity without speeding up. So the actual graph would be curved on either end, but let’s make the same approximation as we did then, anyway.
Looking at this graph, and given what we learned, we can see that there are two distinct periods to the car’s motion—the way to school and the way back. The average velocity for the drive to school is 0.5 km/minute. We can see that the average velocity for the drive back is –0.5 km/minute. If we plot the data showing velocity versus time, we get another graph ():
We can learn a few things. First, we can derive a v versus t graph from a d versus t graph. Second, if we have a straight-line position–time graph that is positively or negatively sloped, it will yield a horizontal velocity graph. There are a few other interesting things to note. Just as we could use a position vs. time graph to determine velocity, we can use a velocity vs. time graph to determine position. We know that v = d/t. If we use a little algebra to re-arrange the equation, we see that d = v t. In , we have velocity on the y-axis and time along the x-axis. Let’s take just the first half of the motion. We get 0.5 km/minute 10 minutes. The units for minutes cancel each other, and we get 5 km, which is the displacement for the trip to school. If we calculate the same for the return trip, we get –5 km. If we add them together, we see that the net displacement for the whole trip is 0 km, which it should be because we started and ended at the same place.
The area under a velocity curve represents the displacement. The velocity curve also tells us whether the car is speeding up. In our earlier example, we stated that the velocity was constant. So, the car is not speeding up. Graphically, you can see that the slope of these two lines is 0. This slope tells us that the car is not speeding up, or accelerating. We will do more with this information in a later chapter. For now, just remember that the area under the graph and the slope are the two important parts of the graph. Just like we could define a linear equation for the motion in a position vs. time graph, we can also define one for a velocity vs. time graph. As we said, the slope equals the acceleration, a. And in this graph, the y-intercept is v0. Thus,
.
But what if the velocity is not constant? Let’s look back at our jet-car example. At the beginning of the motion, as the car is speeding up, we saw that its position is a curve, as shown in .
You do not have to do this, but you could, theoretically, take the instantaneous velocity at each point on this graph. If you did, you would get , which is just a straight line with a positive slope.
Again, if we take the slope of the velocity vs. time graph, we get the acceleration, the rate of change of the velocity. And, if we take the area under the slope, we get back to the displacement.
### Solving Problems using Velocity–Time Graphs
Most velocity vs. time graphs will be straight lines. When this is the case, our calculations are fairly simple.
Most of the velocity vs. time graphs we will look at will be simple to interpret. Occasionally, we will look at curved graphs of velocity vs. time. More often, these curved graphs occur when something is speeding up, often from rest. Let’s look back at a more realistic velocity vs. time graph of the jet car’s motion that takes this speeding up stage into account.
### Practice Problems
### Check Your Understanding
### Section Summary
1. The slope of a velocity vs. time graph is the acceleration.
2. The area under a velocity vs. time curve is the displacement.
3. Average velocity can be found in a velocity vs. time graph by taking the weighted average of all the velocities.
### Key Equations
### Concept Items
### Critical Thinking
### Problems
### Performance Task
### Test Prep Multiple Choice
### Test Prep Short Answer
### Test Prep Extended Response
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# Acceleration
## Introduction
You may have heard the term accelerator, referring to the gas pedal in a car. When the gas pedal is pushed down, the flow of gasoline to the engine increases, which increases the car’s velocity. Pushing on the gas pedal results in acceleration because the velocity of the car increases, and acceleration is defined as a change in velocity. You need two quantities to define velocity: a speed and a direction. Changing either of these quantities, or both together, changes the velocity. You may be surprised to learn that pushing on the brake pedal or turning the steering wheel also causes acceleration. The first reduces the speed and so changes the velocity, and the second changes the direction and also changes the velocity.
In fact, any change in velocity—whether positive, negative, directional, or any combination of these—is called an acceleration in physics. The plane in the picture is said to be accelerating because its velocity is decreasing as it prepares to land. To begin our study of acceleration, we need to have a clear understanding of what acceleration means. |
# Acceleration
## Acceleration
### Section Key Terms
### Defining Acceleration
Throughout this chapter we will use the following terms: time, displacement, velocity, and acceleration. Recall that each of these terms has a designated variable and SI unit of measurement as follows:
1. Time: t, measured in seconds (s)
2. Displacement: Δd, measured in meters (m)
3. Velocity: v, measured in meters per second (m/s)
4. Acceleration: a, measured in meters per second per second (m/s2, also called meters per second squared)
5. Also note the following:
Acceleration is the change in velocity divided by a period of time during which the change occurs. The SI units of velocity are m/s and the SI units for time are s, so the SI units for acceleration are m/s2. Average acceleration is given by
Average acceleration is distinguished from instantaneous acceleration, which is acceleration at a specific instant in time. The magnitude of acceleration is often not constant over time. For example, runners in a race accelerate at a greater rate in the first second of a race than during the following seconds. You do not need to know all the instantaneous accelerations at all times to calculate average acceleration. All you need to know is the change in velocity (i.e., the final velocity minus the initial velocity) and the change in time (i.e., the final time minus the initial time), as shown in the formula. A component of the average acceleration can be positive, negative, or zero. A negative acceleration component is simply an acceleration in the negative direction along that axis. When the motion is in one dimension, we often simply refer to this as negative acceleration, and acceleration in the positive direction as positive acceleration.
Keep in mind that although acceleration points in the same direction as the change in velocity, it is not always in the direction of the velocity itself. When an object slows down, its acceleration is opposite to the direction of its velocity. In everyday language, this is called deceleration; but in physics, it is acceleration—whose direction happens to be opposite that of the velocity. For now, let us assume that motion to the right along the x-axis is positive and motion to the left is negative.
shows a car with positive acceleration in (a) and negative acceleration in (b). The arrows represent vectors showing both direction and magnitude of velocity and acceleration.
Velocity and acceleration are both vector quantities. Recall that vectors have both magnitude and direction. An object traveling at a constant speed does accelerate if it changes direction. So, turning the steering wheel of a moving car makes the car accelerate because the velocity changes direction.
### Calculating Average Acceleration
Look back at the equation for average acceleration. You can see that the calculation of average acceleration involves three values: change in time, (Δt); change in velocity, (Δv); and acceleration (a).
Change in time is often stated as a time interval, and change in velocity can often be calculated by subtracting the initial velocity from the final velocity. Average acceleration is then simply change in velocity divided by change in time. Before you begin calculating, be sure that all distances and times have been converted to meters and seconds. Look at these examples of acceleration of a subway train.
Recall that for analysis of motion in only one dimension, we do not need to use vectors. Instead, we can treat displacement, velocity, and acceleration as scalars. Unlike magnitudes, scalars can be positive or negative, so the sign can serve to tell us the direction along the one-dimensional axis.
### Practice Problems
### Check Your Understanding
### Section Summary
1. Acceleration is the rate of change of velocity. Its magnitude is expressed in units of m/s2.
2. The components of acceleration may be positive or negative.
### Key Equations
### Concept Items
### Critical Thinking
### Problems
### Test Prep Multiple Choice
### Test Prep Short Answer
### Test Prep Extended Response
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# Acceleration
## Representing Acceleration with Equations and Graphs
### Section Key Terms
### How the Kinematic Equations are Related to Acceleration
We are studying concepts related to motion: time, displacement, velocity, and especially acceleration. We are only concerned with motion in one dimension. The kinematic equations that we will be using apply to conditions of constant acceleration, except where noted, and show how these concepts are related. Constant acceleration is acceleration that does not change over time. The first kinematic equation relates displacement d, average velocity
, and time t.
The initial displacement
is often 0, in which case the equation can be written as
This equation, which is the definition of average velocity and valid for both constant and non-constant acceleration, says that average velocity is displacement per unit time. We will express velocity in meters per second. If we graph displacement versus time, as in , the slope will be the velocity. Whenever a rate, such as velocity, is represented graphically, time is usually taken to be the independent variable and is plotted along the x axis.
The second kinematic equation, another expression for average velocity
is simply the initial velocity plus the final velocity divided by two. This equation is only valid for constant acceleration.
Now we come to our main focus of this chapter; namely, the kinematic equations that describe motion with constant acceleration. In the third kinematic equation, acceleration is the rate at which velocity increases, so velocity at any point equals initial velocity plus acceleration multiplied by time
Note that this third kinematic equation does not have displacement in it. Therefore, if you do not know the displacement and are not trying to solve for a displacement, this equation might be a good one to use.
The third kinematic equation is also represented by the graph in .
The fourth kinematic equation shows how displacement is related to acceleration
When starting at the origin, and, when starting from rest, , in which case the equation can be written as
This equation tells us that, for constant acceleration, the slope of a plot of 2d versus t2 is acceleration, as shown in .
The fifth kinematic equation relates velocity, acceleration, and displacement
This equation is useful for when we do not know, or do not need to know, the time.
When starting from rest, the fifth equation simplifies to
According to this equation, a graph of velocity squared versus twice the displacement will have a slope equal to acceleration.
Note that, in reality, knowns and unknowns will vary. Sometimes you will want to rearrange a kinematic equation so that the knowns are the values on the axes and the unknown is the slope. Sometimes the intercept will not be at the origin (0,0). This will happen when v0 or d0 is not zero. This will be the case when the object of interest is already in motion, or the motion begins at some point other than at the origin of the coordinate system.
The kinematic equations are applicable when you have constant acceleration.
1.
, or
when d0 = 0
2.
3.
, or
when v0 = 0
4.
, or
when d0 = 0 and v0 = 0
5.
, or
when d0 = 0 and v0 = 0
### Applying Kinematic Equations to Situations of Constant Acceleration
Problem-solving skills are essential to success in a science and life in general. The ability to apply broad physical principles, which are often represented by equations, to specific situations is a very powerful form of knowledge. It is much more powerful than memorizing a list of facts. Analytical skills and problem-solving abilities can be applied to new situations, whereas a list of facts cannot be made long enough to contain every possible circumstance. Essential analytical skills will be developed by solving problems in this text and will be useful for understanding physics and science in general throughout your life.
### Problem-Solving Steps
While no single step-by-step method works for every problem, the following general procedures facilitate problem solving and make the answers more meaningful. A certain amount of creativity and insight are required as well.
1. Examine the situation to determine which physical principles are involved. It is vital to draw a simple sketch at the outset. Decide which direction is positive and note that on your sketch.
2. Identify the knowns: Make a list of what information is given or can be inferred from the problem statement. Remember, not all given information will be in the form of a number with units in the problem. If something starts from rest, we know the initial velocity is zero. If something stops, we know the final velocity is zero.
3. Identify the unknowns: Decide exactly what needs to be determined in the problem.
4. Find an equation or set of equations that can help you solve the problem. Your list of knowns and unknowns can help here. For example, if time is not needed or not given, then the fifth kinematic equation, which does not include time, could be useful.
5. Insert the knowns along with their units into the appropriate equation and obtain numerical solutions complete with units. This step produces the numerical answer; it also provides a check on units that can help you find errors. If the units of the answer are incorrect, then an error has been made.
6. Check the answer to see if it is reasonable: Does it make sense? This final step is extremely important because the goal of physics is to accurately describe nature. To see if the answer is reasonable, check its magnitude, its sign, and its units. Are the significant figures correct?
### Summary of Problem Solving
1. Determine the knowns and unknowns.
2. Find an equation that expresses the unknown in terms of the knowns. More than one unknown means more than one equation is needed.
3. Solve the equation or equations.
4. Be sure units and significant figures are correct.
5. Check whether the answer is reasonable.
### Practice Problems
### Constant Acceleration
In many cases, acceleration is not uniform because the force acting on the accelerating object is not constant over time. A situation that gives constant acceleration is the acceleration of falling objects. When air resistance is not a factor, all objects near Earth’s surface fall with an acceleration of about 9.80 m/s2. Although this value decreases slightly with increasing altitude, it may be assumed to be essentially constant. The value of 9.80 m/s2 is labeled g and is referred to as acceleration due to gravity. Gravity is the force that causes nonsupported objects to accelerate downward—or, more precisely, toward the center of Earth. The magnitude of this force is called the weight of the object and is given by mg where m is the mass of the object (in kg). In places other than on Earth, such as the Moon or on other planets, g is not 9.80 m/s2, but takes on other values. When using g for the acceleration a in a kinematic equation, it is usually given a negative sign because the acceleration due to gravity is downward.
### Practice Problems
### Check Your Understanding
### Section Summary
1. The kinematic equations show how time, displacement, velocity, and acceleration are related for objects in motion.
2. In general, kinematic problems can be solved by identifying the kinematic equation that expresses the unknown in terms of the knowns.
3. Displacement, velocity, and acceleration may be displayed graphically versus time.
4. The kinematic equations in this section are valid only for constant acceleration.
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# Forces and Newton’s Laws of Motion
## Introduction
Isaac Newton (1642–1727) was a natural philosopher; a great thinker who combined science and philosophy to try to explain the workings of nature on Earth and in the universe. His laws of motion were just one part of the monumental work that has made him legendary. The development of Newton’s laws marks the transition from the Renaissance period of history to the modern era. This transition was characterized by a revolutionary change in the way people thought about the physical universe. Drawing upon earlier work by scientists Galileo Galilei and Johannes Kepler, Newton’s laws of motion allowed motion on Earth and in space to be predicted mathematically. In this chapter you will learn about force as well as Newton’s first, second, and third laws of motion. |
# Forces and Newton’s Laws of Motion
## Force
### Section Key Terms
### Defining Force and Dynamics
Force is the cause of motion, and motion draws our attention. Motion itself can be beautiful, such as a dolphin jumping out of the water, the flight of a bird, or the orbit of a satellite. The study of motion is called kinematics, but kinematics describes only the way objects move—their velocity and their acceleration. Dynamics considers the forces that affect the motion of moving objects and systems. Newton’s laws of motion are the foundation of dynamics. These laws describe the way objects speed up, slow down, stay in motion, and interact with other objects. They are also universal laws: they apply everywhere on Earth as well as in space.
A force pushes or pulls an object. The object being moved by a force could be an inanimate object, a table, or an animate object, a person. The pushing or pulling may be done by a person, or even the gravitational pull of Earth. Forces have different magnitudes and directions; this means that some forces are stronger than others and can act in different directions. For example, a cannon exerts a strong force on the cannonball that is launched into the air. In contrast, a mosquito landing on your arm exerts only a small force on your arm.
When multiple forces act on an object, the forces combine. Adding together all of the forces acting on an object gives the total force, or net force. An external force is a force that acts on an object within the system from outside the system. This type of force is different than an internal force, which acts between two objects that are both within the system. The net external force combines these two definitions; it is the total combined external force. We discuss further details about net force, external force, and net external force in the coming sections.
Like displacements, velocities, and accelerations, forces are vectors that have magnitude and direction. We may represent a force as the sum of two vectors at right angles. These are its one-dimensional components, which we can represent by a signed scalar quantity. When we do so, we must choose a coordinate system and the direction along each axis that will be the positive direction. This choice must be the same for all vectors in the problem—forces, accelerations, etc.
Commonly, in a problem set in a vertical plane, horizontal and vertical axes are chosen as the two primary directions. The usual convention is to make the positive directions right and up. Sometimes a problem concerns a sloping plane, and it may be more convenient to choose axes parallel to the plane and normal to it. In this case, it is usual to take positive in each axis as the directions that tend upward.
Thus, a horizontal force of –3 N means pushing with 3 N to the left.
Unknown values are taken to be in the positive direction, so if your calculation of a vertical force comes out as +5.2 N, then you know it is an upward force. A negative result means the force is downward.
### Free-Body Diagrams and Examples of Forces
For our first example of force, consider an object hanging from a rope. This example gives us the opportunity to introduce a useful tool known as a free-body diagram. A free-body diagram represents the object being acted upon—that is, the free body—as a single point. Only the forces acting on the body (that is, external forces) are shown and are represented by vectors (which are drawn as arrows). These forces are the only ones shown because only external forces acting on the body affect its motion. We can ignore any internal forces within the body because they cancel each other out, as explained in the section on Newton’s third law of motion. Free-body diagrams are very useful for analyzing forces acting on an object.
shows the force of tension in the rope acting in the upward direction, opposite the force of gravity. The forces are indicated in the free-body diagram by an arrow pointing up, representing tension, and another arrow pointing down, representing gravity. In a free-body diagram, the lengths of the arrows show the relative magnitude (or strength) of the forces. Because forces are vectors, they add just like other vectors. Notice that the two arrows have equal lengths in , which means that the forces of tension and weight are of equal magnitude. Because these forces of equal magnitude act in opposite directions, they are perfectly balanced, so they add together to give a net force of zero.
Not all forces are as noticeable as when you push or pull on an object. Some forces act without physical contact, such as the pull of a magnet (in the case of magnetic force) or the gravitational pull of Earth (in the case of gravitational force).
In the next three sections discussing Newton’s laws of motion, we will learn about three specific types of forces: friction, the normal force, and the gravitational force. To analyze situations involving forces, we will create free-body diagrams to organize the framework of the mathematics for each individual situation.
### Section Summary
1. Dynamics is the study of how forces affect the motion of objects and systems.
2. Force is a push or pull that can be defined in terms of various standards. It is a vector and so has both magnitude and direction.
3. External forces are any forces outside of a body that act on the body. A free-body diagram is a drawing of all external forces acting on a body.
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# Forces and Newton’s Laws of Motion
## Newton's First Law of Motion: Inertia
### Section Key Terms
### Newton’s First Law and Friction
Newton’s first law of motion states the following:
1. A body at rest tends to remain at rest.
2. A body in motion tends to remain in motion at a constant velocity unless acted on by a net external force. (Recall that constant velocity means that the body moves in a straight line and at a constant speed.)
At first glance, this law may seem to contradict your everyday experience. You have probably noticed that a moving object will usually slow down and stop unless some effort is made to keep it moving. The key to understanding why, for example, a sliding box slows down (seemingly on its own) is to first understand that a net external force acts on the box to make the box slow down. Without this net external force, the box would continue to slide at a constant velocity (as stated in Newton’s first law of motion). What force acts on the box to slow it down? This force is called friction. Generally, friction is an external force that acts opposite to the direction of relative motion or to prevent slipping. We further identify different types of friction. Kinetic friction is a force by a surface parallel to the surface that opposes the motion of a sliding object and causes it to slow down. When the friction prevents the object from sliding, it is called static friction. Rolling resistance impedes the rolling of a wheel. Drag opposes and slows the motion of an object through a fluid (see ). Think of friction as a resistance to motion that slows things down.
Consider an air hockey table. When the air is turned off, the puck slides only a short distance before friction slows it to a stop. However, when the air is turned on, it lifts the puck slightly, so the puck experiences very little friction as it moves over the surface. With friction almost eliminated, the puck glides along with very little change in speed. On a frictionless surface, the puck would experience no net external force (ignoring air resistance, which is also a form of friction). Additionally, if we know enough about friction, we can accurately predict how quickly objects will slow down.
Now let’s think about another example. A man pushes a box across a floor at constant velocity by applying a force of +50 N. (The positive sign indicates that, by convention, the direction of motion is to the right.) What is the force of friction that opposes the motion? The force of friction must be −50 N. Why? According to Newton’s first law of motion, any object moving at constant velocity has no net external force acting upon it, which means that the sum of the forces acting on the object must be zero. The mathematical way to say that no net external force acts on an object is
or
So if the man applies +50 N of force, then the force of friction must be −50 N for the two forces to add up to zero (that is, for the two forces to cancel each other). Whenever you encounter the phrase at constant velocity, Newton’s first law tells you that the net external force is zero.
The force of friction depends on two factors: the coefficient of friction and the normal force. For any two surfaces that are in contact with one another, the coefficient of friction is a constant that depends on the nature of the surfaces. The normal force is the force exerted by a surface that pushes on an object out, perpendicular to the surface in response to the object’s pushing into the surface. The normal force prevents the object from penetrating the surface. In equation form, the force of friction is
where μ is the coefficient of friction and N is the normal force. (The coefficient of friction is discussed in more detail in another chapter, and the normal force is discussed in more detail in the section Newton's Third Law of Motion.)
Recall from the section on Force that a net external force acts from outside on the object of interest. A more precise definition is that it acts on the system of interest. A system is one or more objects that you choose to study. It is important to define the system at the beginning of a problem to figure out which forces are external and need to be considered, and which are internal and can be ignored.
For example, in (a), two children push a third child in a wagon at a constant velocity. The system of interest is the wagon plus the small child, as shown in part (b) of the figure. The two children behind the wagon exert external forces on this system (F1, F2). Friction f acting on the wheels is another external force acting on the system. Two more external forces act on the system: the weight W of the system pulling down and the normal force N of the ground pushing up. Notice that the wagon is not accelerating vertically, so Newton’s first law tells us that the normal force balances the weight. Because the wagon is moving forward at a constant velocity, the force of friction must have the same strength as the sum of the forces applied by the two children.
### Mass and Inertia
Inertia is the tendency for an object at rest to remain at rest, or for a moving object to remain in motion in a straight line with constant speed. This key property of objects was first described by Galileo. Later, Newton incorporated the concept of inertia into his first law, which is often referred to as the law of inertia.
As we know from experience, some objects have more inertia than others. For example, changing the motion of a large truck is more difficult than changing the motion of a toy truck. In fact, the inertia of an object is proportional to the mass of the object. Mass is a measure of the amount of matter (or stuff) in an object. The quantity or amount of matter in an object is determined by the number and types of atoms the object contains. Unlike weight (which changes if the gravitational force changes), mass does not depend on gravity. The mass of an object is the same on Earth, in orbit, or on the surface of the moon. In practice, it is very difficult to count and identify all of the atoms and molecules in an object, so mass is usually not determined this way. Instead, the mass of an object is determined by comparing it with the standard kilogram. Mass is therefore expressed in kilograms.
### Section Summary
1. Newton’s first law states that a body at rest remains at rest or, if moving, remains in motion in a straight line at a constant speed, unless acted on by a net external force. This law is also known as the law of inertia.
2. Inertia is the tendency of an object at rest to remain at rest or, if moving, to remain in motion at constant velocity. Inertia is related to an object’s mass.
3. Friction is a force that opposes motion and causes an object or system to slow down.
4. Mass is the quantity of matter in a substance.
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# Forces and Newton’s Laws of Motion
## Newton's Second Law of Motion
### Section Key Terms
### Describing Newton’s Second Law of Motion
Newton’s first law considered bodies at rest or bodies in motion at a constant velocity. The other state of motion to consider is when an object is moving with a changing velocity, which means a change in the speed and/or the direction of motion. This type of motion is addressed by Newton’s second law of motion, which states how force causes changes in motion. Newton’s second law of motion is used to calculate what happens in situations involving forces and motion, and it shows the mathematical relationship between force, mass, and . Mathematically, the second law is most often written as
where Fnet (or ∑F) is the net external force, m is the mass of the system, and a is the acceleration. Note that Fnet and ∑F are the same because the net external force is the sum of all the external forces acting on the system.
First, what do we mean by a change in motion? A change in motion is simply a change in velocity: the speed of an object can become slower or faster, the direction in which the object is moving can change, or both of these variables may change. A change in velocity means, by definition, that an acceleration has occurred. Newton’s first law says that only a nonzero net external force can cause a change in motion, so a net external force must cause an acceleration. Note that acceleration can refer to slowing down or to speeding up. Acceleration can also refer to a change in the direction of motion with no change in speed, because acceleration is the change in velocity divided by the time it takes for that change to occur, and velocity is defined by speed and direction.
From the equation
we see that force is directly proportional to both mass and acceleration, which makes sense. To accelerate two objects from rest to the same velocity, you would expect more force to be required to accelerate the more massive object. Likewise, for two objects of the same mass, applying a greater force to one would accelerate it to a greater velocity.
Now, let’s rearrange Newton’s second law to solve for acceleration. We get
In this form, we can see that acceleration is directly proportional to force, which we write as
where the symbol
means proportional to.
This proportionality mathematically states what we just said in words: acceleration is directly proportional to the net external force. When two variables are directly proportional to each other, then if one variable doubles, the other variable must double. Likewise, if one variable is reduced by half, the other variable must also be reduced by half. In general, when one variable is multiplied by a number, the other variable is also multiplied by the same number. It seems reasonable that the acceleration of a system should be directly proportional to and in the same direction as the net external force acting on the system. An object experiences greater acceleration when acted on by a greater force.
It is also clear from the equation
that acceleration is inversely proportional to mass, which we write as
means that if one variable is multiplied by a number, the other variable must be divided by the same number. Now, it also seems reasonable that acceleration should be inversely proportional to the mass of the system. In other words, the larger the mass (the inertia), the smaller the acceleration produced by a given force. This relationship is illustrated in , which shows that a given net external force applied to a basketball produces a much greater acceleration than when applied to a car.
### Applying Newton’s Second Law
Before putting Newton’s second law into action, it is important to consider units.
The equation
is used to define the units of force in terms of the three basic units of mass, length, and time (recall that acceleration has units of length divided by time squared). The SI unit of force is called the newton (abbreviated N) and is the force needed to accelerate a 1-kg system at the rate of 1 m/s2. That is, because
we have
One of the most important applications of Newton’s second law is to calculate weight (also known as the gravitational force), which is usually represented mathematically as W. When people talk about gravity, they don’t always realize that it is an acceleration. When an object is dropped, it accelerates toward the center of Earth. Newton’s second law states that the net external force acting on an object is responsible for the acceleration of the object. If air resistance is negligible, the net external force on a falling object is only the gravitational force (i.e., the weight of the object).
Weight can be represented by a vector because it has a direction. Down is defined as the direction in which gravity pulls, so weight is normally considered a downward force. By using Newton’s second law, we can figure out the equation for weight.
Consider an object with mass m falling toward Earth. It experiences only the force of gravity (i.e., the gravitational force or weight), which is represented by W. Newton’s second law states that
Because the only force acting on the object is the gravitational force, we have
We know that the acceleration of an object due to gravity is g, so we have
Substituting these two expressions into Newton’s second law gives
This is the equation for weight—the gravitational force on a mass m. On Earth,
so the weight (disregarding for now the direction of the weight) of a 1.0-kg object on Earth is
Although most of the world uses newtons as the unit of force, in the United States the most familiar unit of force is the pound (lb), where 1 N = 0.225 lb.
Recall that although gravity acts downward, it can be assigned a positive or negative value, depending on what the positive direction is in your chosen coordinate system. Be sure to take this into consideration when solving problems with weight. When the downward direction is taken to be negative, as is often the case, acceleration due to gravity becomes –g = −9.8 m/s2 and the weight force is –mg.
When the net external force on an object is its weight, we say that it is in freefall. In this case, the only force acting on the object is the force of gravity. On the surface of Earth, when objects fall downward toward Earth, they are never truly in freefall because there is always some upward force due to air resistance that acts on the object (and there is also the buoyancy force of air, which is similar to the buoyancy force in water that keeps boats afloat).
Gravity varies slightly over the surface of Earth, so the weight of an object depends very slightly on its location on Earth. Weight varies dramatically away from Earth’s surface. On the moon, for example, the acceleration due to gravity is only 1.67 m/s2. Because weight depends on the force of gravity, a 1.0-kg mass weighs 9.8 N on Earth and only about 1.7 N on the moon.
It is important to remember that weight and mass are very different, although they are closely related. Mass is the quantity of matter in an object (how much stuff there is, or how hard it is to accelerate it) and does not vary, but weight is the gravitational force on an object and is proportional to the force of gravity. It is easy to confuse the two, because our experience is confined to Earth, and the weight of an object is essentially the same no matter where you are on Earth. Adding to the confusion, the terms mass and weight are often used interchangeably in everyday language; for example, our medical records often show our weight in kilograms, but never in the correct unit of newtons.
### Practice Problems
### Section Summary
1. Acceleration is a change in velocity, meaning a change in speed, direction, or both.
2. An external force acts on a system from outside the system, as opposed to internal forces, which act between components within the system.
3. Newton’s second law of motion states that the acceleration of a system is directly proportional to and in the same direction as the net external force acting on the system, and inversely proportional to the system’s mass.
4. In equation form, Newton’s second law of motion is
or
This is sometimes written as
or
.
5. The weight of an object of mass m is the force of gravity that acts on it. From Newton’s second law, weight is given by
6. If the only force acting on an object is its weight, then the object is in freefall.
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# Forces and Newton’s Laws of Motion
## Newton's Third Law of Motion
### Section Key Terms
### Describing Newton’s Third Law of Motion
If you have ever stubbed your toe, you have noticed that although your toe initiates the impact, the surface that you stub it on exerts a force back on your toe. Although the first thought that crosses your mind is probably “ouch, that hurt” rather than “this is a great example of Newton’s third law,” both statements are true.
This is exactly what happens whenever one object exerts a force on another—each object experiences a force that is the same strength as the force acting on the other object but that acts in the opposite direction. Everyday experiences, such as stubbing a toe or throwing a ball, are all perfect examples of Newton’s third law in action.
Newton’s third law of motion states that whenever a first object exerts a force on a second object, the first object experiences a force equal in magnitude but opposite in direction to the force that it exerts.
Newton’s third law of motion tells us that forces always occur in pairs, and one object cannot exert a force on another without experiencing the same strength force in return. We sometimes refer to these force pairs as action-reaction pairs, where the force exerted is the action, and the force experienced in return is the reaction (although which is which depends on your point of view).
Newton’s third law is useful for figuring out which forces are external to a system. Recall that identifying external forces is important when setting up a problem, because the external forces must be added together to find the net force.
We can see Newton’s third law at work by looking at how people move about. Consider a swimmer pushing off from the side of a pool, as illustrated in . She pushes against the pool wall with her feet and accelerates in the direction opposite to her push. The wall has thus exerted on the swimmer a force of equal magnitude but in the direction opposite that of her push. You might think that two forces of equal magnitude but that act in opposite directions would cancel, but they do not because they act on different systems.
In this case, there are two different systems that we could choose to investigate: the swimmer or the wall. If we choose the swimmer to be the system of interest, as in the figure, then
is an external force on the swimmer and affects her motion. Because acceleration is in the same direction as the net external force, the swimmer moves in the direction of
Because the swimmer is our system (or object of interest) and not the wall, we do not need to consider the force
because it originates from the swimmer rather than acting on the swimmer. Therefore,
does not directly affect the motion of the system and does not cancel
Note that the swimmer pushes in the direction opposite to the direction in which she wants to move.
Other examples of Newton’s third law are easy to find. As a teacher paces in front of a whiteboard, he exerts a force backward on the floor. The floor exerts a reaction force in the forward direction on the teacher that causes him to accelerate forward. Similarly, a car accelerates because the ground pushes forward on the car's wheels in reaction to the car's wheels pushing backward on the ground. You can see evidence of the wheels pushing backward when tires spin on a gravel road and throw rocks backward.
Another example is the force of a baseball as it makes contact with the bat. Helicopters create lift by pushing air down, creating an upward reaction force. Birds fly by exerting force on air in the direction opposite that in which they wish to fly. For example, the wings of a bird force air downward and backward in order to get lift and move forward. An octopus propels itself forward in the water by ejecting water backward through a funnel in its body, which is similar to how a jet ski is propelled. In these examples, the octopus or jet ski push the water backward, and the water, in turn, pushes the octopus or jet ski forward.
### Applying Newton’s Third Law
Forces are classified and given names based on their source, how they are transmitted, or their effects. In previous sections, we discussed the forces called push, weight, and friction. In this section, applying Newton’s third law of motion will allow us to explore three more forces: the normal force, tension, and thrust. However, because we haven’t yet covered vectors in depth, we’ll only consider one-dimensional situations in this chapter. Another chapter will consider forces acting in two dimensions.
The gravitational force (or weight) acts on objects at all times and everywhere on Earth. We know from Newton’s second law that a net force produces an acceleration; so, why is everything not in a constant state of freefall toward the center of Earth? The answer is the normal force. The normal force is the outward force that a surface applies to an object perpendicular to the surface, and it prevents the object from penetrating it. In the case of an object at rest on a horizontal surface, it is the force needed to support the weight of that object. If an object on a flat surface is not accelerating, the net external force is zero, and the normal force has the same magnitude as the weight of the system but acts in the opposite direction. In equation form, we write that
Note that this equation is only true for a horizontal surface.
The word tension comes from the Latin word meaning to stretch. Tension is the force along the length of a flexible connector, such as a string, rope, chain, or cable. Regardless of the type of connector attached to the object of interest, one must remember that the connector can only pull (or exert tension) in the direction parallel to its length. Tension is a pull that acts parallel to the connector, and that acts in opposite directions at the two ends of the connector. This is possible because a flexible connector is simply a long series of action-reaction forces, except at the two ends where outside objects provide one member of the action-reaction forces.
Consider a person holding a mass on a rope, as shown in .
Tension in the rope must equal the weight of the supported mass, as we can prove by using Newton’s second law. If the 5.00 kg mass in the figure is stationary, then its acceleration is zero, so
The only external forces acting on the mass are its weight W and the tension T supplied by the rope. Summing the external forces to find the net force, we obtain
where T and W are the magnitudes of the tension and weight, respectively, and their signs indicate direction, with up being positive. By substituting mg for Fnet and rearranging the equation, the tension equals the weight of the supported mass, just as you would expect
For a 5.00-kg mass (neglecting the mass of the rope), we see that
Another example of Newton’s third law in action is thrust. Rockets move forward by expelling gas backward at a high velocity. This means that the rocket exerts a large force backward on the gas in the rocket combustion chamber, and the gas, in turn, exerts a large force forward on the rocket in response. This reaction force is called thrust.
### Practice Problems
### Section Summary
1. Newton’s third law of motion states that when one body exerts a force on a second body, the first body experiences a force that is equal in magnitude and opposite in direction to the force that it exerts.
2. When an object rests on a surface, the surface applies a force on the object that opposes the weight of the object. This force acts perpendicular to the surface and is called the normal force.
3. The pulling force that acts along a stretched flexible connector, such as a rope or cable, is called tension. When a rope supports the weight of an object at rest, the tension in the rope is equal to the weight of the object.
4. Thrust is a force that pushes an object forward in response to the backward ejection of mass by the object. Rockets and airplanes are pushed forward by thrust.
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# Motion in Two Dimensions
## Introduction
In Chapter 2, we learned to distinguish between vectors and scalars; the difference being that a vector has magnitude and direction, whereas a scalar has only magnitude and possibly sign. We learned how to deal with vectors in physics by working straightforward one-dimensional vector problems, which may be treated mathematically in the same as scalars. In this chapter, we’ll use vectors to expand our understanding of forces and motion into two dimensions. Most real-world physics problems (such as with the game of pool pictured here) are, after all, either two- or three-dimensional problems and physics is most useful when applied to real physical scenarios. We start by learning the practical skills of graphically adding and subtracting vectors (by using drawings) and analytically (with math). Once we’re able to work with two-dimensional vectors, we apply these skills to problems of projectile motion, inclined planes, and harmonic motion. |
# Motion in Two Dimensions
## Vector Addition and Subtraction: Graphical Methods
### Section Key Terms
### The Graphical Method of Vector Addition and Subtraction
Recall that a vector is a quantity that has magnitude and direction. For example, displacement, velocity, acceleration, and force are all vectors. In one-dimensional or straight-line motion, the direction of a vector can be given simply by a plus or minus sign. Motion that is forward, to the right, or upward is usually considered to be positive (+); and motion that is backward, to the left, or downward is usually considered to be negative (−).
In two dimensions, a vector describes motion in two perpendicular directions, such as vertical and horizontal. For vertical and horizontal motion, each vector is made up of vertical and horizontal components. In a one-dimensional problem, one of the components simply has a value of zero. For two-dimensional vectors, we work with vectors by using a frame of reference such as a coordinate system. Just as with one-dimensional vectors, we graphically represent vectors with an arrow having a length proportional to the vector’s magnitude and pointing in the direction that the vector points.
shows a graphical representation of a vector; the total displacement for a person walking in a city. The person first walks nine blocks east and then five blocks north. Her total displacement does not match her path to her final destination. The displacement simply connects her starting point with her ending point using a straight line, which is the shortest distance. We use the notation that a boldface symbol, such as D, stands for a vector. Its magnitude is represented by the symbol in italics, D, and its direction is given by an angle represented by the symbol
Note that her displacement would be the same if she had begun by first walking five blocks north and then walking nine blocks east.
The head-to-tail method is a graphical way to add vectors. The tail of the vector is the starting point of the vector, and the head (or tip) of a vector is the pointed end of the arrow. The following steps describe how to use the head-to-tail method for graphical vector addition.
1. Let the x-axis represent the east-west direction. Using a ruler and protractor, draw an arrow to represent the first vector (nine blocks to the east), as shown in (a).
2. Let the y-axis represent the north-south direction. Draw an arrow to represent the second vector (five blocks to the north). Place the tail of the second vector at the head of the first vector, as shown in (b).
3. If there are more than two vectors, continue to add the vectors head-to-tail as described in step 2. In this example, we have only two vectors, so we have finished placing arrows tip to tail.
4. Draw an arrow from the tail of the first vector to the head of the last vector, as shown in (c). This is the resultant, or the sum, of the vectors.
5. To find the magnitude of the resultant, measure its length with a ruler. When we deal with vectors analytically in the next section, the magnitude will be calculated by using the Pythagorean theorem.
6. To find the direction of the resultant, use a protractor to measure the angle it makes with the reference direction (in this case, the x-axis). When we deal with vectors analytically in the next section, the direction will be calculated by using trigonometry to find the angle.
Vector subtraction is done in the same way as vector addition with one small change. We add the first vector to the negative of the vector that needs to be subtracted. A negative vector has the same magnitude as the original vector, but points in the opposite direction (as shown in ). Subtracting the vector B from the vector A, which is written as A − B, is the same as A + (−B). Since it does not matter in what order vectors are added, A − B is also equal to (−B) + A. This is true for scalars as well as vectors. For example, 5 – 2 = 5 + (−2) = (−2) + 5.
Global angles are calculated in the counterclockwise direction. The clockwise direction is considered negative. For example, an angle of
south of west is the same as the global angle
which can also be expressed as
from the positive x-axis.
### Using the Graphical Method of Vector Addition and Subtraction to Solve Physics Problems
Now that we have the skills to work with vectors in two dimensions, we can apply vector addition to graphically determine the resultant vector, which represents the total force. Consider an example of force involving two ice skaters pushing a third as seen in .
In problems where variables such as force are already known, the forces can be represented by making the length of the vectors proportional to the magnitudes of the forces. For this, you need to create a scale. For example, each centimeter of vector length could represent 50 N worth of force. Once you have the initial vectors drawn to scale, you can then use the head-to-tail method to draw the resultant vector. The length of the resultant can then be measured and converted back to the original units using the scale you created.
You can tell by looking at the vectors in the free-body diagram in that the two skaters are pushing on the third skater with equal-magnitude forces, since the length of their force vectors are the same. Note, however, that the forces are not equal because they act in different directions. If, for example, each force had a magnitude of 400 N, then we would find the magnitude of the total external force acting on the third skater by finding the magnitude of the resultant vector. Since the forces act at a right angle to one another, we can use the Pythagorean theorem. For a triangle with sides a, b, and c, the Pythagorean theorem tells us that
Applying this theorem to the triangle made by F1, F2, and Ftot in , we get
or
Note that, if the vectors were not at a right angle to each other
to one another), we would not be able to use the Pythagorean theorem to find the magnitude of the resultant vector. Another scenario where adding two-dimensional vectors is necessary is for velocity, where the direction may not be purely east-west or north-south, but some combination of these two directions. In the next section, we cover how to solve this type of problem analytically. For now let’s consider the problem graphically.
### Practice Problems
### Section Summary
1. The graphical method of adding vectors
and
involves drawing vectors on a graph and adding them by using the head-to-tail method. The resultant vector
is defined such that A + B = R. The magnitude and direction of
are then determined with a ruler and protractor.
2. The graphical method of subtracting vectors A and B involves adding the opposite of vector B, which is defined as −B. In this case,
Next, use the head-to-tail method as for vector addition to obtain the resultant vector
.
3. Addition of vectors is independent of the order in which they are added; A + B = B + A.
4. The head-to-tail method of adding vectors involves drawing the first vector on a graph and then placing the tail of each subsequent vector at the head of the previous vector. The resultant vector is then drawn from the tail of the first vector to the head of the final vector.
5. Variables in physics problems, such as force or velocity, can be represented with vectors by making the length of the vector proportional to the magnitude of the force or velocity.
6. Problems involving displacement, force, or velocity may be solved graphically by measuring the resultant vector’s magnitude with a ruler and measuring the direction with a protractor.
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# Motion in Two Dimensions
## Vector Addition and Subtraction: Analytical Methods
### Section Key Terms
### Components of Vectors
For the analytical method of vector addition and subtraction, we use some simple geometry and trigonometry, instead of using a ruler and protractor as we did for graphical methods. However, the graphical method will still come in handy to visualize the problem by drawing vectors using the head-to-tail method. The analytical method is more accurate than the graphical method, which is limited by the precision of the drawing. For a refresher on the definitions of the sine, cosine, and tangent of an angle, see .
Since, by definition, , we can find the length x if we know h and by using . Similarly, we can find the length of y by using . These trigonometric relationships are useful for adding vectors.
When a vector acts in more than one dimension, it is useful to break it down into its x and y components. For a two-dimensional vector, a component is a piece of a vector that points in either the x- or y-direction. Every 2-d vector can be expressed as a sum of its x and y components.
For example, given a vector like
in , we may want to find what two perpendicular vectors,
and
, add to produce it. In this example,
and
form a right triangle, meaning that the angle between them is 90 degrees. This is a common situation in physics and happens to be the least complicated situation trigonometrically.
and
are defined to be the components of
along the x- and y-axes. The three vectors,
,
, and
, form a right triangle.
If the vector
is known, then its magnitude
(its length) and its angle
(its direction) are known. To find
and
, its x- and y-components, we use the following relationships for a right triangle:
and
where
is the magnitude of A in the x-direction,
is the magnitude of A in the y-direction, and
is the angle of the resultant with respect to the x-axis, as shown in .
Suppose, for example, that
is the vector representing the total displacement of the person walking in a city, as illustrated in .
Then A = 10.3 blocks and
, so that
This magnitude indicates that the walker has traveled 9 blocks to the east—in other words, a 9-block eastward displacement. Similarly,
indicating that the walker has traveled 5 blocks to the north—a 5-block northward displacement.
### Analytical Method of Vector Addition and Subtraction
Calculating a resultant vector (or vector addition) is the reverse of breaking the resultant down into its components. If the perpendicular components
and
of a vector
are known, then we can find
analytically. How do we do this? Since, by definition,
we solve for
to find the direction of the resultant.
Note that gives an angle in the first quadrant if
and in the fourth quadrant if
. If, in fact, both and are negative, or if is negative and positive, then , measured from the positive direction, is
.
Since this is a right triangle, the Pythagorean theorem (x2 + y2 = h2) for finding the hypotenuse applies. In this case, it becomes
Solving for A gives
In summary, to find the magnitude
and direction
of a vector from its perpendicular components
and
, as illustrated in , we use the following relationships:
Sometimes, the vectors added are not perfectly perpendicular to one another. An example of this is the case below, where the vectors
and
are added to produce the resultant
as illustrated in .
If
and
represent two legs of a walk (two displacements), then
is the total displacement. The person taking the walk ends up at the tip of
. There are many ways to arrive at the same point. The person could have walked straight ahead first in the x-direction and then in the y-direction. Those paths are the x- and y-components of the resultant,
and
If we know
and
, we can find
and
using the equations
and
.
1. Draw in the x and y components of each vector (including the resultant) with a dashed line. Use the equations
and
to find the components. In , these components are
,
,
, and
Vector
makes an angle of
with the x-axis, and vector
makes and angle of
with its own x-axis (which is slightly above the x-axis used by vector A).
2. Find the x component of the resultant by adding the x component of the vectors
and find the
Now that we know the components of
3. To get the magnitude of the resultant R, use the Pythagorean theorem.
4. To get the direction of the resultant
### Using the Analytical Method of Vector Addition and Subtraction to Solve Problems
uses the analytical method to add vectors.
### Practice Problems
### Section Summary
1. The analytical method of vector addition and subtraction uses the Pythagorean theorem and trigonometric identities to determine the magnitude and direction of a resultant vector.
2. The steps to add vectors
and
using the analytical method are as follows:
### Key Equations
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# Motion in Two Dimensions
## Projectile Motion
### Section Key Terms
### Properties of Projectile Motion
Projectile motion is the motion of an object thrown (projected) into the air when, after the initial force that launches the object, air resistance is negligible and the only other force that object experiences is the force of gravity. The object is called a projectile, and its path is called its trajectory. Air resistance is a frictional force that slows its motion and can significantly alter the trajectory of the motion. Due to the difficulty in calculation, only situations in which the deviation from projectile motion is negligible and air resistance can be ignored are considered in introductory physics. That approximation is often quite accurate.
The most important concept in projectile motion is that when air resistance is ignored, horizontal and vertical motions are , meaning that they don’t influence one another. compares a cannonball in free fall (in blue) to a cannonball launched horizontally in projectile motion (in red). You can see that the cannonball in free fall falls at the same rate as the cannonball in projectile motion. Keep in mind that if the cannon launched the ball with any vertical component to the velocity, the vertical displacements would not line up perfectly.
Since vertical and horizontal motions are independent, we can analyze them separately, along perpendicular axes. To do this, we separate projectile motion into the two components of its motion, one along the horizontal axis and the other along the vertical.
We’ll call the horizontal axis the x-axis and the vertical axis the y-axis. For notation, d is the total displacement, and x and y are its components along the horizontal and vertical axes. The magnitudes of these vectors are x and y, as illustrated in .
As usual, we use velocity, acceleration, and displacement to describe motion. We must also find the components of these variables along the x- and y-axes. The components of acceleration are then very simple a = –g = –9.80 m/s2. Note that this definition defines the upwards direction as positive. Because gravity is vertical, a = 0. Both accelerations are constant, so we can use the kinematic equations. For review, the kinematic equations from a previous chapter are summarized in .
Where x is position, x0 is initial position, v is velocity, vavg is average velocity, t is time and a is acceleration.
### Solve Problems Involving Projectile Motion
The following steps are used to analyze projectile motion:
1. Separate the motion into horizontal and vertical components along the x- and y-axes. These axes are perpendicular, so
and
are used. The magnitudes of the displacement
along x- and y-axes are called
and
The magnitudes of the components of the velocity
are
and
, where
is the magnitude of the velocity and
is its direction. Initial values are denoted with a subscript 0.
2.
Treat the motion as two independent one-dimensional motions, one horizontal and the other vertical. The kinematic equations for horizontal and vertical motion take the following forms
Vertical motion (assuming positive is up
)
3. Solve for the unknowns in the two separate motions (one horizontal and one vertical). Note that the only common variable between the motions is time
. The problem solving procedures here are the same as for one-dimensional kinematics.
4. Recombine the two motions to find the total displacement
and velocity
. We can use the analytical method of vector addition, which uses
and
to find the magnitude and direction of the total displacement and velocity.
is the direction of the displacement
, and
is the direction of the velocity
. (See
The expression we found for
while solving part (a) of the previous problem works for any projectile motion problem where air resistance is negligible. Call the maximum height
; then,
This equation defines the maximum height of a projectile. The maximum height depends only on the vertical component of the initial velocity.
### Practice Problems
The fact that vertical and horizontal motions are independent of each other lets us predict the range of a projectile. The range is the horizontal distance R traveled by a projectile on level ground, as illustrated in . Throughout history, people have been interested in finding the range of projectiles for practical purposes, such as aiming cannons.
How does the initial velocity of a projectile affect its range? Obviously, the greater the initial speed
, the greater the range, as shown in the figure above. The initial angle
also has a dramatic effect on the range. When air resistance is negligible, the range
of a projectile on level ground is
where
is the initial speed and
is the initial angle relative to the horizontal. It is important to note that the range doesn’t apply to problems where the initial and final y position are different, or to cases where the object is launched perfectly horizontally.
### Section Summary
1. Projectile motion is the motion of an object through the air that is subject only to the acceleration of gravity.
2. Projectile motion in the horizontal and vertical directions are independent of one another.
3. The maximum height of an projectile is the highest altitude, or maximum displacement in the vertical position reached in the path of a projectile.
4. The range is the maximum horizontal distance traveled by a projectile.
5. To solve projectile problems: choose a coordinate system; analyze the motion in the vertical and horizontal direction separately; then, recombine the horizontal and vertical components using vector addition equations.
### Key Equations
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# Motion in Two Dimensions
## Inclined Planes
### Section Key Terms
### Static Friction and Kinetic Friction
Recall from the previous chapter that friction is a force that opposes relative motion parallel to the contact surface of the interacting objects and is around us all the time. Friction allows us to move, which you have discovered if you have ever tried to walk on ice.
There are different types of friction—kinetic and static. Kinetic friction acts on an object in relative motion, while static friction acts on an object or system at rest relative to each other. The maximum static friction is usually greater than the kinetic friction between the objects.
Imagine, for example, trying to slide a heavy crate across a concrete floor. You may push harder and harder on the crate and not move it at all. This means that the static friction responds to what you do—it increases to be equal to and in the opposite direction of your push. But if you finally push hard enough, the crate seems to slip suddenly and starts to move. Once in motion, it is easier to keep it in motion than it was to get it started because the kinetic friction force is less than the static friction force. If you were to add mass to the crate, (for example, by placing a box on top of it) you would need to push even harder to get it started and also to keep it moving. If, on the other hand, you oiled the concrete you would find it easier to get the crate started and keep it going.
shows how friction occurs at the interface between two objects. Magnifying these surfaces shows that they are rough on the microscopic level. So when you push to get an object moving (in this case, a crate), you must raise the object until it can skip along with just the tips of the surface hitting, break off the points, or do both. The harder the surfaces are pushed together (such as if another box is placed on the crate), the more force is needed to move them.
The magnitude of the frictional force has two forms: one for static friction, the other for kinetic friction. When there is no motion between the objects, the magnitude of static friction f is
where
is the coefficient of static friction and N is the magnitude of the normal force. Recall that the normal force acts perpendicular to the surface and prevents the crate from falling through the floor. It opposes the force of gravity in this example, but that will not always be the case.
Since the symbol
means less than or equal to, this equation says that static friction can have a maximum value of
That is,
Static friction is a responsive force that increases to be equal and opposite to whatever force is exerted, up to its maximum limit. Once the applied force exceeds fs(max), the object will move. Once an object is moving, the magnitude of kinetic friction fk is given by
where
is the coefficient of kinetic friction.
Friction varies from surface to surface because different substances are rougher than others. compares values of static and kinetic friction for different surfaces. The coefficient of the friction depends on the two surfaces that are in contact.
Since friction always opposes relative motion, the direction of friction is upward along the plane if the object is at rest or sliding down the incline. For example, if the crate you try to push (with a force parallel to the floor) has a mass of 100 kg, then the normal force would be equal to its weight
perpendicular to the floor. If the coefficient of static friction is 0.45, you would have to exert a force parallel to the floor greater than
to move the crate. Once there is motion, friction is less and the coefficient of kinetic friction might be 0.30, so that a force of only 290 N
would keep it moving at a constant speed. If the floor were lubricated, both coefficients would be much smaller than they would be without lubrication. The coefficient of friction is unitless and is a number usually between 0 and 1.0, but there is no theoretical upper limit to its value.
### Working with Inclined Planes
We discussed previously that when an object rests on a horizontal surface, there is a normal force supporting it equal in magnitude to its weight. Up until now, we dealt only with normal force in one dimension, with gravity and normal force acting perpendicular to the surface in opposing directions (gravity downward, and normal force upward). Now that you have the skills to work with forces in two dimensions, we can explore what happens to weight and the normal force on a tilted surface such as an inclined plane. For inclined plane problems, it is easier breaking down the forces into their components if we rotate the coordinate system, as illustrated in . The first step when setting up the problem is to break down the force of weight into components.
When an object rests on an incline that makes an angle
with the horizontal, the force of gravity acting on the object is divided into two components: A force acting perpendicular to the plane,
, and a force acting parallel to the plane,
. The perpendicular force of weight,
, is typically equal in magnitude and opposite in direction to the normal force,
The force acting parallel to the plane,
, causes the object to accelerate down the incline. The force of friction,
, opposes the motion of the object, so it acts upward along the plane.
It is important to be careful when resolving the weight of the object into components. If the angle of the incline is at an angle
to the horizontal, then the magnitudes of the weight components are
Instead of memorizing these equations, it is helpful to be able to determine them from reason. To do this, draw the right triangle formed by the three weight vectors. Notice that the angle of the incline is the same as the angle formed between
and
. Knowing this property, you can use trigonometry to determine the magnitude of the weight components
To review, the process for solving inclined plane problems is as follows:
1. Draw a sketch of the problem.
2. Identify known and unknown quantities, and identify the system of interest.
3. Draw a free-body diagram (which is a sketch showing all of the forces acting on an object) with the coordinate system rotated at the same angle as the inclined plane. Resolve the vectors into horizontal and vertical components and draw them on the free-body diagram.
4. Write Newton’s second law in the horizontal and vertical directions and add the forces acting on the object. If the object does not accelerate in a particular direction (for example, the x -direction) then Fnet x = 0. If the object does accelerate in that direction, Fnet x = ma.
5. Check your answer. Is the answer reasonable? Are the units correct?
### Practice Problems
### Section Summary
1. Friction is a contact force between systems that opposes the motion or attempted motion between them. Simple friction is proportional to the normal force N pushing the systems together. A normal force is always perpendicular to the contact surface between systems. Friction depends on both of the materials involved.
2. µs is the coefficient of static friction, which depends on both of the materials.
3. µk is the coefficient of kinetic friction, which also depends on both materials.
4. When objects rest on an inclined plane that makes an angle
with the horizontal surface, the weight of the object can be broken into components that act perpendicular
and parallel (
) to the surface of the plane.
### Key Equations
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# Motion in Two Dimensions
## Simple Harmonic Motion
### Section Key Terms
### Hooke’s Law and Simple Harmonic Motion
Imagine a car parked against a wall. If a bulldozer pushes the car into the wall, the car will not move but it will noticeably change shape. A change in shape due to the application of a force is a deformation. Even very small forces are known to cause some deformation. For small deformations, two important things can happen. First, unlike the car and bulldozer example, the object returns to its original shape when the force is removed. Second, the size of the deformation is proportional to the force. This second property is known as Hooke’s law. In equation form, Hooke’s law is
where x is the amount of deformation (the change in length, for example) produced by the restoring force F, and k is a constant that depends on the shape and composition of the object. The restoring force is the force that brings the object back to its equilibrium position; the minus sign is there because the restoring force acts in the direction opposite to the displacement. Note that the restoring force is proportional to the deformation x. The deformation can also be thought of as a displacement from equilibrium. It is a change in position due to a force. In the absence of force, the object would rest at its equilibrium position. The force constant k is related to the stiffness of a system. The larger the force constant, the stiffer the system. A stiffer system is more difficult to deform and requires a greater restoring force. The units of k are newtons per meter (N/m). One of the most common uses of Hooke’s law is solving problems involving springs and pendulums, which we will cover at the end of this section.
### Oscillations and Periodic Motion
What do an ocean buoy, a child in a swing, a guitar, and the beating of hearts all have in common? They all oscillate. That is, they move back and forth between two points, like the ruler illustrated in . All oscillations involve force. For example, you push a child in a swing to get the motion started.
Newton’s first law implies that an object oscillating back and forth is experiencing forces. Without force, the object would move in a straight line at a constant speed rather than oscillate. Consider, for example, plucking a plastic ruler to the left as shown in . The deformation of the ruler creates a force in the opposite direction, known as a restoring force. Once released, the restoring force causes the ruler to move back toward its stable equilibrium position, where the net force on it is zero. However, by the time the ruler gets there, it gains momentum and continues to move to the right, producing the opposite deformation. It is then forced to the left, back through equilibrium, and the process is repeated until it gradually loses all of its energy. The simplest oscillations occur when the restoring force is directly proportional to displacement. Recall that Hooke’s law describes this situation with the equation F = −kx. Therefore, Hooke’s law describes and applies to the simplest case of oscillation, known as simple harmonic motion.
When you pluck a guitar string, the resulting sound has a steady tone and lasts a long time. Each vibration of the string takes the same time as the previous one. Periodic motion is a motion that repeats itself at regular time intervals, such as with an object bobbing up and down on a spring or a pendulum swinging back and forth. The time to complete one oscillation (a complete cycle of motion) remains constant and is called the period T. Its units are usually seconds.
Frequency f is the number of oscillations per unit time. The SI unit for frequency is the hertz (Hz), defined as the number of oscillations per second. The relationship between frequency and period is
As you can see from the equation, frequency and period are different ways of expressing the same concept. For example, if you get a paycheck twice a month, you could say that the frequency of payment is two per month, or that the period between checks is half a month.
If there is no friction to slow it down, then an object in simple motion will oscillate forever with equal displacement on either side of the equilibrium position. The equilibrium position is where the object would naturally rest in the absence of force. The maximum displacement from equilibrium is called the amplitude X. The units for amplitude and displacement are the same, but depend on the type of oscillation. For the object on the spring, shown in , the units of amplitude and displacement are meters.
The mass m and the force constant k are the only factors that affect the period and frequency of simple harmonic motion. The period of a simple harmonic oscillator is given by
and, because f = 1/T, the frequency of a simple harmonic oscillator is
### Solving Spring and Pendulum Problems with Simple Harmonic Motion
Before solving problems with springs and pendulums, it is important to first get an understanding of how a pendulum works. provides a useful illustration of a simple pendulum.
Everyday examples of pendulums include old-fashioned clocks, a child’s swing, or the sinker on a fishing line. For small displacements of less than 15 degrees, a pendulum experiences simple harmonic oscillation, meaning that its restoring force is directly proportional to its displacement. A pendulum in simple harmonic motion is called a simple pendulum. A pendulum has an object with a small mass, also known as the pendulum bob, which hangs from a light wire or string. The equilibrium position for a pendulum is where the angle
is zero (that is, when the pendulum is hanging straight down). It makes sense that without any force applied, this is where the pendulum bob would rest.
The displacement of the pendulum bob is the arc length s. The weight mg has components mg cos
along the string and mg sin
tangent to the arc. Tension in the string exactly cancels the component mg cos
parallel to the string. This leaves a net restoring force back toward the equilibrium position that runs tangent to the arc and equals −mg sin
.
For small angle oscillations of a simple pendulum, the period is
The only things that affect the period of a simple pendulum are its length and the acceleration due to gravity. The period is completely independent of other factors, such as mass or amplitude. However, note that T does depend on g. This means that if we know the length of a pendulum, we can actually use it to measure gravity! This will come in useful in .
### Practice Problems
### Section Summary
1. An oscillation is a back and forth motion of an object between two points of deformation.
2. An oscillation may create a wave, which is a disturbance that propagates from where it was created.
3. The simplest type of oscillations are related to systems that can be described by Hooke’s law.
4. Periodic motion is a repetitious oscillation.
5. The time for one oscillation is the period T.
6. The number of oscillations per unit time is the frequency
7. A mass m suspended by a wire of length L is a simple pendulum and undergoes simple harmonic motion for amplitudes less than about 15 degrees.
### Key Equations
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# Circular and Rotational Motion
## Introduction
You may recall learning about various aspects of motion along a straight line: kinematics (where we learned about displacement, velocity, and acceleration), projectile motion (a special case of two-dimensional kinematics), force, and Newton’s laws of motion. In some ways, this chapter is a continuation of Newton’s laws of motion. Recall that Newton’s first law tells us that objects move along a straight line at constant speed unless a net external force acts on them. Therefore, if an object moves along a circular path, such as the car in the photo, it must be experiencing an external force. In this chapter, we explore both circular motion and rotational motion. |
# Circular and Rotational Motion
## Angle of Rotation and Angular Velocity
### Section Key Terms
### Angle of Rotation
What exactly do we mean by circular motion or rotation? Rotational motion is the circular motion of an object about an axis of rotation. We will discuss specifically circular motion and spin. Circular motion is when an object moves in a circular path. Examples of circular motion include a race car speeding around a circular curve, a toy attached to a string swinging in a circle around your head, or the circular loop-the-loop on a roller coaster. Spin is rotation about an axis that goes through the center of mass of the object, such as Earth rotating on its axis, a wheel turning on its axle, the spin of a tornado on its path of destruction, or a figure skater spinning during a performance at the Olympics. Sometimes, objects will be spinning while in circular motion, like the Earth spinning on its axis while revolving around the Sun, but we will focus on these two motions separately.
When solving problems involving rotational motion, we use variables that are similar to linear variables (distance, velocity, acceleration, and force) but take into account the curvature or rotation of the motion. Here, we define the angle of rotation, which is the angular equivalence of distance; and angular velocity, which is the angular equivalence of linear velocity.
When objects rotate about some axis—for example, when the CD in rotates about its center—each point in the object follows a circular path.
The arc length, Δ, is the distance traveled along a circular path. The radius of curvature, , is the radius of the circular path. Both are shown in .
Consider a line from the center of the CD to its edge. In a given time, each pit (used to record information) on this line moves through the same angle. The angle of rotation is the amount of rotation and is the angular analog of distance. The angle of rotation
is the arc length divided by the radius of curvature.
The angle of rotation is often measured by using a unit called the radian. (Radians are actually dimensionless, because a radian is defined as the ratio of two distances, radius and arc length.) A revolution is one complete rotation, where every point on the circle returns to its original position. One revolution covers
radians (or 360 degrees), and therefore has an angle of rotation of
radians, and an arc length that is the same as the circumference of the circle. We can convert between radians, revolutions, and degrees using the relationship
1 revolution =
rad = 360°. See for the conversion of degrees to radians for some common angles.
### Angular Velocity
How fast is an object rotating? We can answer this question by using the concept of angular velocity. Consider first the angular speed
is the rate at which the angle of rotation changes. In equation form, the angular speed is
which means that an angular rotation
occurs in a time,
. If an object rotates through a greater angle of rotation in a given time, it has a greater angular speed. The units for angular speed are radians per second (rad/s).
Now let’s consider the direction of the angular speed, which means we now must call it the angular velocity. The direction of the angular velocity is along the axis of rotation. For an object rotating clockwise, the angular velocity points away from you along the axis of rotation. For an object rotating counterclockwise, the angular velocity points toward you along the axis of rotation.
Angular velocity (ω) is the angular version of linear velocity v. Tangential velocity is the instantaneous linear velocity of an object in rotational motion. To get the precise relationship between angular velocity and tangential velocity, consider again a pit on the rotating CD. This pit moves through an arc length
in a short time
so its tangential speed is
From the definition of the angle of rotation,
, we see that
. Substituting this into the expression for v gives
The equation
says that the tangential speed v is proportional to the distance r from the center of rotation. Consequently, tangential speed is greater for a point on the outer edge of the CD (with larger r) than for a point closer to the center of the CD (with smaller r). This makes sense because a point farther out from the center has to cover a longer arc length in the same amount of time as a point closer to the center. Note that both points will still have the same angular speed, regardless of their distance from the center of rotation. See .
Now, consider another example: the tire of a moving car (see ). The faster the tire spins, the faster the car moves—large
means large v because
. Similarly, a larger-radius tire rotating at the same angular velocity,
, will produce a greater linear (tangential) velocity, v, for the car. This is because a larger radius means a longer arc length must contact the road, so the car must move farther in the same amount of time.
However, there are cases where linear velocity and tangential velocity are not equivalent, such as a car spinning its tires on ice. In this case, the linear velocity will be less than the tangential velocity. Due to the lack of friction under the tires of a car on ice, the arc length through which the tire treads move is greater than the linear distance through which the car moves. It’s similar to running on a treadmill or pedaling a stationary bike; you are literally going nowhere fast.
### Solving Problems Involving Angle of Rotation and Angular Velocity
Now that we have an understanding of the concepts of angle of rotation and angular velocity, we’ll apply them to the real-world situations of a clock tower and a spinning tire.
### Practice Problems
### Check Your Understanding
### Section Summary
1. Circular motion is motion in a circular path.
2. The angle of rotation
is defined as the ratio of the arc length to the radius of curvature.
3. The arc length
is the distance traveled along a circular path and r is the radius of curvature of the circular path.
4. The angle of rotation
is measured in units of radians (rad), where
revolution.
5. Angular velocity
is the rate of change of an angle, where a rotation
occurs in a time
.
6. The units of angular velocity are radians per second (rad/s).
7. Tangential speed v and angular speed
are related by
, and tangential velocity has units of m/s.
8. The direction of angular velocity is along the axis of rotation, away (toward) from you for clockwise (counterclockwise) motion.
### Key Equations
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# Circular and Rotational Motion
## Uniform Circular Motion
### Section Key Terms
### Centripetal Acceleration
In the previous section, we defined circular motion. The simplest case of circular motion is uniform circular motion, where an object travels a circular path at a constant speed. Note that, unlike speed, the linear velocity of an object in circular motion is constantly changing because it is always changing direction. We know from kinematics that acceleration is a change in velocity, either in magnitude or in direction or both. Therefore, an object undergoing uniform circular motion is always accelerating, even though the magnitude of its velocity is constant.
You experience this acceleration yourself every time you ride in a car while it turns a corner. If you hold the steering wheel steady during the turn and move at a constant speed, you are executing uniform circular motion. What you notice is a feeling of sliding (or being flung, depending on the speed) away from the center of the turn. This isn’t an actual force that is acting on you—it only happens because your body wants to continue moving in a straight line (as per Newton’s first law) whereas the car is turning off this straight-line path. Inside the car it appears as if you are forced away from the center of the turn. This fictitious force is known as the centrifugal force. The sharper the curve and the greater your speed, the more noticeable this effect becomes.
shows an object moving in a circular path at constant speed. The direction of the instantaneous tangential velocity is shown at two points along the path. Acceleration is in the direction of the change in velocity; in this case it points roughly toward the center of rotation. (The center of rotation is at the center of the circular path). If we imagine
becoming smaller and smaller, then the acceleration would point exactly toward the center of rotation, but this case is hard to draw. We call the acceleration of an object moving in uniform circular motion the centripetal acceleration ac because centripetal means center seeking.
Now that we know that the direction of centripetal acceleration is toward the center of rotation, let’s discuss the magnitude of centripetal acceleration. For an object traveling at speed v in a circular path with radius r, the magnitude of centripetal acceleration is
Centripetal acceleration is greater at high speeds and in sharp curves (smaller radius), as you may have noticed when driving a car, because the car actually pushes you toward the center of the turn. But it is a bit surprising that ac is proportional to the speed squared. This means, for example, that the acceleration is four times greater when you take a curve at 100 km/h than at 50 km/h.
We can also express ac in terms of the magnitude of angular velocity. Substituting
into the equation above, we get
. Therefore, the magnitude of centripetal acceleration in terms of the magnitude of angular velocity is
### Centripetal Force
Because an object in uniform circular motion undergoes acceleration (by changing the direction of motion but not the speed), we know from Newton’s second law of motion that there must be a net external force acting on the object. Since the magnitude of the acceleration is constant, so is the magnitude of the net force, and since the acceleration points toward the center of the rotation, so does the net force.
Any force or combination of forces can cause a centripetal acceleration. Just a few examples are the tension in the rope on a tether ball, the force of Earth’s gravity on the Moon, the friction between a road and the tires of a car as it goes around a curve, or the normal force of a roller coaster track on the cart during a loop-the-loop.
The component of any net force that causes circular motion is called a centripetal force. When the net force is equal to the centripetal force, and its magnitude is constant, uniform circular motion results. The direction of a centripetal force is toward the center of rotation, the same as for centripetal acceleration. According to Newton’s second law of motion, a net force causes the acceleration of mass according to Fnet = ma. For uniform circular motion, the acceleration is centripetal acceleration: a = ac. Therefore, the magnitude of centripetal force, Fc, is
.
By using the two different forms of the equation for the magnitude of centripetal acceleration,
and
, we get two expressions involving the magnitude of the centripetal force . The first expression is in terms of tangential speed, the second is in terms of angular speed:
and
.
Both forms of the equation depend on mass, velocity, and the radius of the circular path. You may use whichever expression for centripetal force is more convenient. Newton’s second law also states that the object will accelerate in the same direction as the net force. By definition, the centripetal force is directed towards the center of rotation, so the object will also accelerate towards the center. A straight line drawn from the circular path to the center of the circle will always be perpendicular to the tangential velocity. Note that, if you solve the first expression for r, you get
From this expression, we see that, for a given mass and velocity, a large centripetal force causes a small radius of curvature—that is, a tight curve.
### Solving Centripetal Acceleration and Centripetal Force Problems
To get a feel for the typical magnitudes of centripetal acceleration, we’ll do a lab estimating the centripetal acceleration of a tennis racket and then, in our first Worked Example, compare the centripetal acceleration of a car rounding a curve to gravitational acceleration. For the second Worked Example, we’ll calculate the force required to make a car round a curve.
### Practice Problems
### Check Your Understanding
### Section Summary
1. Centripetal acceleration ac is the acceleration experienced while in uniform circular motion.
2. Centripetal acceleration force is a center-seeking force that always points toward the center of rotation, perpendicular to the linear velocity, in the same direction as the net force, and in the direction opposite that of the radius vector.
3. The standard unit for centripetal acceleration is m/s2.
4. Centripetal force Fc is any net force causing uniform circular motion.
### Key Equations
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# Circular and Rotational Motion
## Rotational Motion
### Section Key Terms
### Rotational Kinematics
In the section on uniform circular motion, we discussed motion in a circle at constant speed and, therefore, constant angular velocity. However, there are times when angular velocity is not constant—rotational motion can speed up, slow down, or reverse directions. Angular velocity is not constant when a spinning skater pulls in her arms, when a child pushes a merry-go-round to make it rotate, or when a CD slows to a halt when switched off. In all these cases, angular acceleration occurs because the angular velocity
changes. The faster the change occurs, the greater is the angular acceleration. Angular acceleration
is the rate of change of angular velocity. In equation form, average angular acceleration is
where
is the change in angular velocity and
is the change in time. The units of angular acceleration are (rad/s)/s, or rad/s2. If
increases, then
is positive. If
decreases, then
is negative. Keep in mind that, by convention, counterclockwise is the positive direction and clockwise is the negative direction. For example, the skater in is rotating counterclockwise as seen from above, so her angular velocity is positive. Acceleration would be negative, for example, when an object that is rotating counterclockwise slows down. It would be positive when an object that is rotating counterclockwise speeds up.
The relationship between the magnitudes of tangential acceleration, a, and angular acceleration,
These equations mean that the magnitudes of tangential acceleration and angular acceleration are directly proportional to each other. The greater the angular acceleration, the larger the change in tangential acceleration, and vice versa. For example, consider riders in their pods on a Ferris wheel at rest. A Ferris wheel with greater angular acceleration will give the riders greater tangential acceleration because, as the Ferris wheel increases its rate of spinning, it also increases its tangential velocity. Note that the radius of the spinning object also matters. For example, for a given angular acceleration
, a smaller Ferris wheel leads to a smaller tangential acceleration for the riders.
So far, we have defined three rotational variables:
,
, and
. These are the angular versions of the linear variables x, v, and a. The following equations in the table represent the magnitude of the rotational variables and only when the radius is constant and perpendicular to the rotational variable. shows how they are related.
We can now begin to see how rotational quantities like
,
, and
are related to each other. For example, if a motorcycle wheel that starts at rest has a large angular acceleration for a fairly long time, it ends up spinning rapidly and rotates through many revolutions. Putting this in terms of the variables, if the wheel’s angular acceleration
is large for a long period of time t, then the final angular velocity
and angle of rotation
are large. In the case of linear motion, if an object starts at rest and undergoes a large linear acceleration, then it has a large final velocity and will have traveled a large distance.
The kinematics of rotational motion describes the relationships between the angle of rotation, angular velocity, angular acceleration, and time. It only describes motion—it does not include any forces or masses that may affect rotation (these are part of dynamics). Recall the kinematics equation for linear motion:
(constant a).
As in linear kinematics, we assume a is constant, which means that angular acceleration
is also a constant, because
. The equation for the kinematics relationship between
,
, and t is
where
is the initial angular velocity. Notice that the equation is identical to the linear version, except with angular analogs of the linear variables. In fact, all of the linear kinematics equations have rotational analogs, which are given in . These equations can be used to solve rotational or linear kinematics problem in which a and
are constant.
In these equations,
and
are initial values,
is zero, and the average angular velocity
and average velocity
are
### Torque
If you have ever spun a bike wheel or pushed a merry-go-round, you know that force is needed to change angular velocity. The farther the force is applied from the pivot point (or fulcrum), the greater the angular acceleration. For example, a door opens slowly if you push too close to its hinge, but opens easily if you push far from the hinges. Furthermore, we know that the more massive the door is, the more slowly it opens; this is because angular acceleration is inversely proportional to mass. These relationships are very similar to the relationships between force, mass, and acceleration from Newton’s second law of motion. Since we have already covered the angular versions of distance, velocity and time, you may wonder what the angular version of force is, and how it relates to linear force.
The angular version of force is torque
, which is the turning effectiveness of a force. See . The equation for the magnitude of torque is
where r is the magnitude of the lever arm, F is the magnitude of the linear force, and
is the angle between the lever arm and the force. The lever arm is the vector from the point of rotation (pivot point or fulcrum) to the location where force is applied. Since the magnitude of the lever arm is a distance, its units are in meters, and torque has units of N⋅m. Torque is a vector quantity and has the same direction as the angular acceleration that it produces.
Applying a stronger torque will produce a greater angular acceleration. For example, the harder the man pushes the merry-go-round in , the faster it accelerates. Furthermore, the more massive the merry-go-round is, the slower it accelerates for the same torque. If the man wants to maximize the effect of his force on the merry-go-round, he should push as far from the center as possible to get the largest lever arm and, therefore, the greatest torque and angular acceleration. Torque is also maximized when the force is applied perpendicular to the lever arm.
### Solving Rotational Kinematics and Torque Problems
Just as linear forces can balance to produce zero net force and no linear acceleration, the same is true of rotational motion. When two torques of equal magnitude act in opposing directions, there is no net torque and no angular acceleration, as you can see in the following video. If zero net torque acts on a system spinning at a constant angular velocity, the system will continue to spin at the same angular velocity.
Now let’s look at examples applying rotational kinematics to a fishing reel and the concept of torque to a merry-go-round.
### Practice Problems
### Check Your Understanding
### Section Summary
1. Kinematics is the description of motion.
2. The kinematics of rotational motion describes the relationships between rotation angle, angular velocity, angular acceleration, and time.
3. Torque is the effectiveness of a force to change the rotational speed of an object. Torque is the rotational analog of force.
4. The lever arm is the distance between the point of rotation (pivot point) and the location where force is applied.
5. Torque is maximized by applying force perpendicular to the lever arm and at a point as far as possible from the pivot point or fulcrum. If torque is zero, angular acceleration is zero.
### Key Equations
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# Newton's Law of Gravitation
## Introduction
What do a falling apple and the orbit of the moon have in common? You will learn in this chapter that each is caused by gravitational force. The motion of all celestial objects, in fact, is determined by the gravitational force, which depends on their mass and separation.
Johannes Kepler discovered three laws of planetary motion that all orbiting planets and moons follow. Years later, Isaac Newton found these laws useful in developing his law of universal gravitation. This law relates gravitational force to the masses of objects and the distance between them. Many years later still, Albert Einstein showed there was a little more to the gravitation story when he published his theory of general relativity. |
# Newton's Law of Gravitation
## Kepler's Laws of Planetary Motion
### Section Key Terms
### Concepts Related to Kepler’s Laws of Planetary Motion
Examples of orbits abound. Hundreds of artificial satellites orbit Earth together with thousands of pieces of debris. The moon’s orbit around Earth has intrigued humans from time immemorial. The orbits of planets, asteroids, meteors, and comets around the sun are no less interesting. If we look farther, we see almost unimaginable numbers of stars, galaxies, and other celestial objects orbiting one another and interacting through gravity.
All these motions are governed by gravitational force. The orbital motions of objects in our own solar system are simple enough to describe with a few fairly simple laws. The orbits of planets and moons satisfy the following two conditions:
1. The mass of the orbiting object, m, is small compared to the mass of the object it orbits, M.
2. The system is isolated from other massive objects.
Based on the motion of the planets about the sun, Kepler devised a set of three classical laws, called Kepler’s laws of planetary motion, that describe the orbits of all bodies satisfying these two conditions:
1. The orbit of each planet around the sun is an ellipse with the sun at one focus.
2. Each planet moves so that an imaginary line drawn from the sun to the planet sweeps out equal areas in equal times.
3. The ratio of the squares of the periods of any two planets about the sun is equal to the ratio of the cubes of their average distances from the sun.
These descriptive laws are named for the German astronomer Johannes Kepler (1571–1630). He devised them after careful study (over some 20 years) of a large amount of meticulously recorded observations of planetary motion done by Tycho Brahe (1546–1601). Such careful collection and detailed recording of methods and data are hallmarks of good science. Data constitute the evidence from which new interpretations and meanings can be constructed. Let’s look closer at each of these laws.
### Kepler’s First Law
The orbit of each planet about the sun is an ellipse with the sun at one focus, as shown in . The planet’s closest approach to the sun is called perihelion and its farthest distance from the sun is called aphelion.
If you know the aphelion (ra) and perihelion (rp) distances, then you can calculate the semi-major axis (a) and semi-minor axis (b).
### Kepler’s Second Law
Each planet moves so that an imaginary line drawn from the sun to the planet sweeps out equal areas in equal times, as shown in .
### Kepler’s Third Law
The ratio of the periods squared of any two planets around the sun is equal to the ratio of their average distances from the sun cubed. In equation form, this is
where T is the period (time for one orbit) and r is the average distance (also called orbital radius). This equation is valid only for comparing two small masses orbiting a single large mass. Most importantly, this is only a descriptive equation; it gives no information about the cause of the equality.
### Calculations Related to Kepler’s Laws of Planetary Motion
### Kepler’s First Law
Refer back to (a). Notice which distances are constant. The foci are fixed, so distance
is a constant. The definition of an ellipse states that the sum of the distances
is also constant. These two facts taken together mean that the perimeter of triangle
must also be constant. Knowledge of these constants will help you determine positions and distances of objects in a system that includes one object orbiting another.
### Kepler’s Second Law
Refer back to . The second law says that the segments have equal area and that it takes equal time to sweep through each segment. That is, the time it takes to travel from A to B equals the time it takes to travel from C to D, and so forth. Velocity v equals distance d divided by time t:
. Then,
, so distance divided by velocity is also a constant. For example, if we know the average velocity of Earth on June 21 and December 21, we can compare the distance Earth travels on those days.
The degree of elongation of an elliptical orbit is called its eccentricity (e). Eccentricity is calculated by dividing the distance f from the center of an ellipse to one of the foci by half the long axis a.
When
, the ellipse is a circle.
The area of an ellipse is given by
, where b is half the short axis. If you know the axes of Earth’s orbit and the area Earth sweeps out in a given period of time, you can calculate the fraction of the year that has elapsed.
### Kepler’s Third Law
Kepler’s third law states that the ratio of the squares of the periods of any two planets (T1, T2) is equal to the ratio of the cubes of their average orbital distance from the sun (r1, r2). Mathematically, this is represented by
From this equation, it follows that the ratio r3/T2 is the same for all planets in the solar system. Later we will see how the work of Newton leads to a value for this constant.
### Practice Problems
### Check Your Understanding
### Section Summary
1. All satellites follow elliptical orbits.
2. The line from the satellite to the parent body sweeps out equal areas in equal time.
3. The radius cubed divided by the period squared is a constant for all satellites orbiting the same parent body.
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# Newton's Law of Gravitation
## Newton's Law of Universal Gravitation and Einstein's Theory of General Relativity
### Section Key Terms
### Concepts Related to Newton’s Law of Universal Gravitation
Sir Isaac Newton was the first scientist to precisely define the gravitational force, and to show that it could explain both falling bodies and astronomical motions. See . But Newton was not the first to suspect that the same force caused both our weight and the motion of planets. His forerunner, Galileo Galilei, had contended that falling bodies and planetary motions had the same cause. Some of Newton’s contemporaries, such as Robert Hooke, Christopher Wren, and Edmund Halley, had also made some progress toward understanding gravitation. But Newton was the first to propose an exact mathematical form and to use that form to show that the motion of heavenly bodies should be conic sections—circles, ellipses, parabolas, and hyperbolas. This theoretical prediction was a major triumph. It had been known for some time that moons, planets, and comets follow such paths, but no one had been able to propose an explanation of the mechanism that caused them to follow these paths and not others.
The gravitational force is relatively simple. It is always attractive, and it depends only on the masses involved and the distance between them. Expressed in modern language, Newton’s universal law of gravitation states that every object in the universe attracts every other object with a force that is directed along a line joining them. The force is directly proportional to the product of their masses and inversely proportional to the square of the distance between them. This attraction is illustrated by .
For two bodies having masses m and M with a distance r between their centers of mass, the equation for Newton’s universal law of gravitation is
where F is the magnitude of the gravitational force and G is a proportionality factor called the gravitational constant. G is a universal constant, meaning that it is thought to be the same everywhere in the universe. It has been measured experimentally to be
.
If a person has a mass of 60.0 kg, what would be the force of gravitational attraction on him at Earth’s surface? G is given above, Earth’s mass M is 5.97 × 1024 kg, and the radius r of Earth is 6.38 × 106 m. Putting these values into Newton’s universal law of gravitation gives
We can check this result with the relationship:
You may remember that g, the acceleration due to gravity, is another important constant related to gravity. By substituting g for a in the equation for Newton’s second law of motion we get
. Combining this with the equation for universal gravitation gives
Cancelling the mass m on both sides of the equation and filling in the values for the gravitational constant and mass and radius of the Earth, gives the value of g, which may look familiar.
This is a good point to recall the difference between mass and weight. Mass is the amount of matter in an object; weight is the force of attraction between the mass within two objects. Weight can change because g is different on every moon and planet. An object’s mass m does not change but its weight mg can.
It is possible to derive Kepler’s third law from Newton’s law of universal gravitation. Applying Newton’s second law of motion to angular motion gives an expression for centripetal force, which can be equated to the expression for force in the universal gravitation equation. This expression can be manipulated to produce the equation for Kepler’s third law. We saw earlier that the expression r is a constant for satellites orbiting the same massive object. The derivation of Kepler’s third law from Newton’s law of universal gravitation and Newton’s second law of motion yields that constant:
where M is the mass of the central body about which the satellites orbit (for example, the sun in our solar system). The usefulness of this equation will be seen later.
The universal gravitational constant G is determined experimentally. This determination was first done accurately in 1798 by English scientist Henry Cavendish (1731–1810), more than 100 years after Newton published his universal law of gravitation. The measurement of G is very basic and important because it determines the strength of one of the four forces in nature. Cavendish’s experiment was very difficult because he measured the tiny gravitational attraction between two ordinary-sized masses (tens of kilograms at most) by using an apparatus like that in . Remarkably, his value for G differs by less than 1% from the modern value.
### Einstein’s Theory of General Relativity
Einstein’s theory of general relativity explained some interesting properties of gravity not covered by Newton’s theory. Einstein based his theory on the postulate that acceleration and gravity have the same effect and cannot be distinguished from each other. He concluded that light must fall in both a gravitational field and in an accelerating reference frame. shows this effect (greatly exaggerated) in an accelerating elevator. In (a), the elevator accelerates upward in zero gravity. In (b), the room is not accelerating but is subject to gravity. The effect on light is the same: it “falls” downward in both situations. The person in the elevator cannot tell whether the elevator is accelerating in zero gravity or is stationary and subject to gravity. Thus, gravity affects the path of light, even though we think of gravity as acting between masses, while photons are massless.
Einstein’s theory of general relativity got its first verification in 1919 when starlight passing near the sun was observed during a solar eclipse. (See .) During an eclipse, the sky is darkened and we can briefly see stars. Those on a line of sight nearest the sun should have a shift in their apparent positions. Not only was this shift observed, but it agreed with Einstein’s predictions well within experimental uncertainties. This discovery created a scientific and public sensation. Einstein was now a folk hero as well as a very great scientist. The bending of light by matter is equivalent to a bending of space itself, with light following the curve. This is another radical change in our concept of space and time. It is also another connection that any particle with mass or energy (e.g., massless photons) is affected by gravity.
To summarize the two views of gravity, Newton envisioned gravity as a tug of war along the line connecting any two objects in the universe. In contrast, Einstein envisioned gravity as a bending of space-time by mass.
### Calculations Based on Newton’s Law of Universal Gravitation
The mass of an object is constant, but its weight varies with the strength of the gravitational field. This means the value of g varies from place to place in the universe. The relationship between force, mass, and acceleration from the second law of motion can be written in terms of g.
In this case, the force is the weight of the object, which is caused by the gravitational attraction of the planet or moon on which the object is located. We can use this expression to compare weights of an object on different moons and planets.
Two equations involving the gravitational constant, G, are often useful. The first is Newton’s equation,
. Several of the values in this equation are either constants or easily obtainable. F is often the weight of an object on the surface of a large object with mass M, which is usually known. The mass of the smaller object, m, is often known, and G is a universal constant with the same value anywhere in the universe. This equation can be used to solve problems involving an object on or orbiting Earth or other massive celestial object. Sometimes it is helpful to equate the right-hand side of the equation to mg and cancel the m on both sides.
The equation
is also useful for problems involving objects in orbit. Note that there is no need to know the mass of the object. Often, we know the radius r or the period T and want to find the other. If these are both known, we can use the equation to calculate the mass of a planet or star.
### Practice Problems
### Check Your Understanding
### Section Summary
1. Newton’s law of universal gravitation provides a mathematical basis for gravitational force and Kepler’s laws of planetary motion.
2. Einstein’s theory of general relativity shows that gravitational fields change the path of light and warp space and time.
3. An object’s mass is constant, but its weight changes when acceleration due to gravity, g, changes.
### Key Equations
### Concept Items
### Critical Thinking
### Performance Task
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# Momentum
## Introduction
We know from everyday use of the word momentum that it is a tendency to continue on course in the same direction. Newscasters speak of sports teams or politicians gaining, losing, or maintaining the momentum to win. As we learned when studying about inertia, which is Newton's first law of motion, every object or system has inertia—that is, a tendency for an object in motion to remain in motion or an object at rest to remain at rest. Mass is a useful variable that lets us quantify inertia. Momentum is mass in motion.
Momentum is important because it is conserved in isolated systems; this fact is convenient for solving problems where objects collide. The magnitude of momentum grows with greater mass and/or speed. For example, look at the football players in the photograph (). They collide and fall to the ground. During their collisions, momentum will play a large part. In this chapter, we will learn about momentum, the different types of collisions, and how to use momentum equations to solve collision problems. |
# Momentum
## Linear Momentum, Force, and Impulse
### Section Key Terms
### Momentum, Impulse, and the Impulse-Momentum Theorem
Linear momentum is the product of a system’s mass and its velocity. In equation form, linear momentum p is
You can see from the equation that momentum is directly proportional to the object’s mass (m) and velocity (v). Therefore, the greater an object’s mass or the greater its velocity, the greater its momentum. A large, fast-moving object has greater momentum than a smaller, slower object.
Momentum is a vector and has the same direction as velocity v. Since mass is a scalar, when velocity is in a negative direction (i.e., opposite the direction of motion), the momentum will also be in a negative direction; and when velocity is in a positive direction, momentum will likewise be in a positive direction. The SI unit for momentum is kg m/s.
Momentum is so important for understanding motion that it was called the quantity of motion by physicists such as Newton. Force influences momentum, and we can rearrange Newton’s second law of motion to show the relationship between force and momentum.
Recall our study of Newton’s second law of motion (Fnet = ma). Newton actually stated his second law of motion in terms of momentum: The net external force equals the change in momentum of a system divided by the time over which it changes. The change in momentum is the difference between the final and initial values of momentum.
In equation form, this law is
where Fnet is the net external force,
is the change in momentum, and
is the change in time.
We can solve for
by rearranging the equation
to be
is known as impulse and this equation is known as the impulse-momentum theorem. From the equation, we see that the impulse equals the average net external force multiplied by the time this force acts. It is equal to the change in momentum. The effect of a force on an object depends on how long it acts, as well as the strength of the force. Impulse is a useful concept because it quantifies the effect of a force. A very large force acting for a short time can have a great effect on the momentum of an object, such as the force of a racket hitting a tennis ball. A small force could cause the same change in momentum, but it would have to act for a much longer time.
### Newton’s Second Law in Terms of Momentum
When Newton’s second law is expressed in terms of momentum, it can be used for solving problems where mass varies, since
. In the more traditional form of the law that you are used to working with, mass is assumed to be constant. In fact, this traditional form is a special case of the law, where mass is constant.
is actually derived from the equation:
For the sake of understanding the relationship between Newton’s second law in its two forms, let’s recreate the derivation of
from
by substituting the definitions of acceleration and momentum.
The change in momentum
is given by
If the mass of the system is constant, then
By substituting
for
, Newton’s second law of motion becomes
for a constant mass.
Because
we can substitute to get the familiar equation
when the mass of the system is constant.
### Solving Problems Using the Impulse-Momentum Theorem
### Practice Problems
### Check Your Understanding
### Section Summary
1. Linear momentum, often referenced as momentum for short, is defined as the product of a system’s mass multiplied by its velocity, p = mv.
2. The SI unit for momentum is kg m/s.
3. Newton’s second law of motion in terms of momentum states that the net external force equals the change in momentum of a system divided by the time over which it changes,
.
4. Impulse is the average net external force multiplied by the time this force acts, and impulse equals the change in momentum,
.
5. Forces are usually not constant over a period of time, so we use the average of the force over the time it acts.
### Key Equations
### Concept Items
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# Momentum
## Conservation of Momentum
### Section Key Terms
### Conservation of Momentum
It is important we realize that momentum is conserved during collisions, explosions, and other events involving objects in motion. To say that a quantity is conserved means that it is constant throughout the event. In the case of conservation of momentum, the total momentum in the system remains the same before and after the collision.
You may have noticed that momentum was not conserved in some of the examples previously presented in this chapter. where forces acting on the objects produced large changes in momentum. Why is this? The systems of interest considered in those problems were not inclusive enough. If the systems were expanded to include more objects, then momentum would in fact be conserved in those sample problems. It is always possible to find a larger system where momentum is conserved, even though momentum changes for individual objects within the system.
For example, if a football player runs into the goalpost in the end zone, a force will cause him to bounce backward. His momentum is obviously greatly changed, and considering only the football player, we would find that momentum is not conserved. However, the system can be expanded to contain the entire Earth. Surprisingly, Earth also recoils—conserving momentum—because of the force applied to it through the goalpost. The effect on Earth is not noticeable because it is so much more massive than the player, but the effect is real.
Next, consider what happens if the masses of two colliding objects are more similar than the masses of a football player and Earth—in the example shown in of one car bumping into another. Both cars are coasting in the same direction when the lead car, labeled m2, is bumped by the trailing car, labeled m1. The only unbalanced force on each car is the force of the collision, assuming that the effects due to friction are negligible. Car m1 slows down as a result of the collision, losing some momentum, while car m2 speeds up and gains some momentum. If we choose the system to include both cars and assume that friction is negligible, then the momentum of the two-car system should remain constant. Now we will prove that the total momentum of the two-car system does in fact remain constant, and is therefore conserved.
Using the impulse-momentum theorem, the change in momentum of car 1 is given by
where F1 is the force on car 1 due to car 2, and
is the time the force acts, or the duration of the collision.
Similarly, the change in momentum of car 2 is
where F2 is the force on car 2 due to car 1, and we assume the duration of the collision
is the same for both cars. We know from Newton’s third law of motion that F2 = –F1, and so
.
Therefore, the changes in momentum are equal and opposite, and
.
Because the changes in momentum add to zero, the total momentum of the two-car system is constant. That is,
where p′1 and p′2 are the momenta of cars 1 and 2 after the collision.
This result that momentum is conserved is true not only for this example involving the two cars, but for any system where the net external force is zero, which is known as an isolated system. The law of conservation of momentum states that for an isolated system with any number of objects in it, the total momentum is conserved. In equation form, the law of conservation of momentum for an isolated system is written as
or
where ptot is the total momentum, or the sum of the momenta of the individual objects in the system at a given time, and p′tot is the total momentum some time later.
The conservation of momentum principle can be applied to systems as diverse as a comet striking the Earth or a gas containing huge numbers of atoms and molecules. Conservation of momentum appears to be violated only when the net external force is not zero. But another larger system can always be considered in which momentum is conserved by simply including the source of the external force. For example, in the collision of two cars considered above, the two-car system conserves momentum while each one-car system does not.
### Check Your Understanding
### Section Summary
1. The law of conservation of momentum is written ptot = constant or ptot = p′tot (isolated system), where ptot is the initial total momentum and p′tot is the total momentum some time later.
2. In an isolated system, the net external force is zero.
3. Conservation of momentum applies only when the net external force is zero, within the defined system.
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# Momentum
## Elastic and Inelastic Collisions
### Section Key Terms
### Elastic and Inelastic Collisions
When objects collide, they can either stick together or bounce off one another, remaining separate. In this section, we’ll cover these two different types of collisions, first in one dimension and then in two dimensions.
In an elastic collision, the objects separate after impact and don’t lose any of their kinetic energy. Kinetic energy is the energy of motion and is covered in detail elsewhere. The law of conservation of momentum is very useful here, and it can be used whenever the net external force on a system is zero. shows an elastic collision where momentum is conserved.
An animation of an elastic collision between balls can be seen by watching this video. It replicates the elastic collisions between balls of varying masses.
Perfectly elastic collisions can happen only with subatomic particles. Everyday observable examples of perfectly elastic collisions don’t exist—some kinetic energy is always lost, as it is converted into heat transfer due to friction. However, collisions between everyday objects are almost perfectly elastic when they occur with objects and surfaces that are nearly frictionless, such as with two steel blocks on ice.
Now, to solve problems involving one-dimensional elastic collisions between two objects, we can use the equation for conservation of momentum. First, the equation for conservation of momentum for two objects in a one-dimensional collision is
Substituting the definition of momentum p = mv for each initial and final momentum, we get
where the primes (') indicate values after the collision; In some texts, you may see i for initial (before collision) and f for final (after collision). The equation assumes that the mass of each object does not change during the collision.
Now, let us turn to the second type of collision. An inelastic collision is one in which kinetic energy is not conserved. A perfectly inelastic collision (also sometimes called completely or maximally inelastic) is one in which objects stick together after impact, and the maximum amount of kinetic energy is lost. This lack of conservation means that the forces between colliding objects may convert kinetic energy to other forms of energy, such as potential energy or thermal energy. The concepts of energy are discussed more thoroughly elsewhere. For inelastic collisions, kinetic energy may be lost in the form of heat. shows an example of an inelastic collision. Two objects that have equal masses head toward each other at equal speeds and then stick together. The two objects come to rest after sticking together, conserving momentum but not kinetic energy after they collide. Some of the energy of motion gets converted to thermal energy, or heat.
Since the two objects stick together after colliding, they move together at the same speed. This lets us simplify the conservation of momentum equation from
to
for inelastic collisions, where v′ is the final velocity for both objects as they are stuck together, either in motion or at rest.
### Solving Collision Problems
The Khan Academy videos referenced in this section show examples of elastic and inelastic collisions in one dimension. In one-dimensional collisions, the incoming and outgoing velocities are all along the same line. But what about collisions, such as those between billiard balls, in which objects scatter to the side? These are two-dimensional collisions, and just as we did with two-dimensional forces, we will solve these problems by first choosing a coordinate system and separating the motion into its x and y components.
One complication with two-dimensional collisions is that the objects might rotate before or after their collision. For example, if two ice skaters hook arms as they pass each other, they will spin in circles. We will not consider such rotation until later, and so for now, we arrange things so that no rotation is possible. To avoid rotation, we consider only the scattering of point masses—that is, structureless particles that cannot rotate or spin.
We start by assuming that Fnet = 0, so that momentum p is conserved. The simplest collision is one in which one of the particles is initially at rest. The best choice for a coordinate system is one with an axis parallel to the velocity of the incoming particle, as shown in . Because momentum is conserved, the components of momentum along the x- and y-axes, displayed as pand p, will also be conserved. With the chosen coordinate system, pis initially zero and pis the momentum of the incoming particle.
Now, we will take the conservation of momentum equation, p1 + p2 = p′1 + p′2 and break it into its x and y components.
Along the x-axis, the equation for conservation of momentum is
In terms of masses and velocities, this equation is
But because particle 2 is initially at rest, this equation becomes
The components of the velocities along the x-axis have the form v cos θ . Because particle 1 initially moves along the x-axis, we find v1= v1. Conservation of momentum along the x-axis gives the equation
where
and
are as shown in .
Along the y-axis, the equation for conservation of momentum is
or
But v1y is zero, because particle 1 initially moves along the x-axis. Because particle 2 is initially at rest, v2y is also zero. The equation for conservation of momentum along the y-axis becomes
The components of the velocities along the y-axis have the form v sin
. Therefore, conservation of momentum along the y-axis gives the following equation:
### Practice Problems
### Check Your Understanding
### Section Summary
1. If kinetic energy is conserved, the collision is elastic. If they stick together, the collision is perfectly inelastic.
2. Kinetic energy is conserved in an elastic collision, but not in an inelastic collision.
3. The approach to two-dimensional collisions is to choose a convenient coordinate system and break the motion into components along perpendicular axes. Choose a coordinate system with the x-axis parallel to the velocity of the incoming particle.
4. Two-dimensional collisions of point masses, where mass 2 is initially at rest, conserve momentum along the initial direction of mass 1, or the x-axis, and along the direction perpendicular to the initial direction, or the y-axis.
5. Point masses are structureless particles that cannot spin.
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# Work, Energy, and Simple Machines
## Introduction
Roller coasters have provided thrills for daring riders around the world since the nineteenth century. Inventors of roller coasters used simple physics to build the earliest examples using railroad tracks on mountainsides and old mines. Modern roller coaster designers use the same basic laws of physics to create the latest amusement park favorites. Physics principles are used to engineer the machines that do the work to lift a roller coaster car up its first big incline before it is set loose to roll. Engineers also have to understand the changes in the car’s energy that keep it speeding over hills, through twists, turns, and even loops.
What exactly is energy? How can changes in force, energy, and simple machines move objects like roller coaster cars? How can machines help us do work? In this chapter, you will discover the answer to this question and many more, as you learn about work, energy, and simple machines. |
# Work, Energy, and Simple Machines
## Work, Power, and the Work–Energy Theorem
### Section Key Terms
### The Work–Energy Theorem
In physics, the term work has a very specific definition. Work is application of force, , to move an object over a distance, d, in the direction that the force is applied. Work, W, is described by the equation
Some things that we typically consider to be work are not work in the scientific sense of the term. Let’s consider a few examples. Think about why each of the following statements is true.
1. Homework is not work.
2. Lifting a rock upwards off the ground is work.
3. Carrying a rock in a straight path across the lawn at a constant speed is not work.
The first two examples are fairly simple. Homework is not work because objects are not being moved over a distance. Lifting a rock up off the ground is work because the rock is moving in the direction that force is applied. The last example is less obvious. Recall from the laws of motion that force is not required to move an object at constant velocity. Therefore, while some force may be applied to keep the rock up off the ground, no net force is applied to keep the rock moving forward at constant velocity.
Work and energy are closely related. When you do work to move an object, you change the object’s energy. You (or an object) also expend energy to do work. In fact, energy can be defined as the ability to do work. Energy can take a variety of different forms, and one form of energy can transform to another. In this chapter we will be concerned with mechanical energy, which comes in two forms: kinetic energy and potential energy.
1. Kinetic energy is also called energy of motion. A moving object has kinetic energy.
2. Potential energy, sometimes called stored energy, comes in several forms. Gravitational potential energy is the stored energy an object has as a result of its position above Earth’s surface (or another object in space). A roller coaster car at the top of a hill has gravitational potential energy.
Let’s examine how doing work on an object changes the object’s energy. If we apply force to lift a rock off the ground, we increase the rock’s potential energy, PE. If we drop the rock, the force of gravity increases the rock’s kinetic energy as the rock moves downward until it hits the ground.
The force we exert to lift the rock is equal to its weight, w, which is equal to its mass, m, multiplied by acceleration due to gravity, g.
The work we do on the rock equals the force we exert multiplied by the distance, d, that we lift the rock. The work we do on the rock also equals the rock’s gain in gravitational potential energy, PE.
Kinetic energy depends on the mass of an object and its velocity, v.
When we drop the rock the force of gravity causes the rock to fall, giving the rock kinetic energy. When work done on an object increases only its kinetic energy, then the net work equals the change in the value of the quantity
. This is a statement of the work–energy theorem, which is expressed mathematically as
The subscripts 2 and 1 indicate the final and initial velocity, respectively. This theorem was proposed and successfully tested by James Joule, shown in .
Does the name Joule sound familiar? The joule (J) is the metric unit of measurement for both work and energy. The measurement of work and energy with the same unit reinforces the idea that work and energy are related and can be converted into one another. 1.0 J = 1.0 N∙m, the units of force multiplied by distance. 1.0 N = 1.0 kg∙m/s2, so 1.0 J = 1.0 kg∙m2/s2. Analyzing the units of the term (1/2)mv2 will produce the same units for joules.
### Calculations Involving Work and Power
In applications that involve work, we are often interested in how fast the work is done. For example, in roller coaster design, the amount of time it takes to lift a roller coaster car to the top of the first hill is an important consideration. Taking a half hour on the ascent will surely irritate riders and decrease ticket sales. Let’s take a look at how to calculate the time it takes to do work.
Recall that a rate can be used to describe a quantity, such as work, over a period of time. Power is the rate at which work is done. In this case, rate means per unit of time. Power is calculated by dividing the work done by the time it took to do the work.
Let’s consider an example that can help illustrate the differences among work, force, and power. Suppose the woman in lifting the TV with a pulley gets the TV to the fourth floor in two minutes, and the man carrying the TV up the stairs takes five minutes to arrive at the same place. They have done the same amount of work
on the TV, because they have moved the same mass over the same vertical distance, which requires the same amount of upward force. However, the woman using the pulley has generated more power. This is because she did the work in a shorter amount of time, so the denominator of the power formula, t, is smaller. (For simplicity’s sake, we will leave aside for now the fact that the man climbing the stairs has also done work on himself.)
Power can be expressed in units of watts (W). This unit can be used to measure power related to any form of energy or work. You have most likely heard the term used in relation to electrical devices, especially light bulbs. Multiplying power by time gives the amount of energy. Electricity is sold in kilowatt-hours because that equals the amount of electrical energy consumed.
The watt unit was named after James Watt (1736–1819) (see ). He was a Scottish engineer and inventor who discovered how to coax more power out of steam engines.
Before proceeding, be sure you understand the distinctions among force, work, energy, and power. Force exerted on an object over a distance does work. Work can increase energy, and energy can do work. Power is the rate at which work is done.
### Practice Problems
### Check Your Understanding
### Section Summary
1. Doing work on a system or object changes its energy.
2. The work–energy theorem states that an amount of work that changes the velocity of an object is equal to the change in kinetic energy of that object.The work–energy theorem states that an amount of work that changes the velocity of an object is equal to the change in kinetic energy of that object.
3. Power is the rate at which work is done.
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# Work, Energy, and Simple Machines
## Mechanical Energy and Conservation of Energy
### Section Key Terms
### Mechanical Energy and Conservation of Energy
We saw earlier that mechanical energy can be either potential or kinetic. In this section we will see how energy is transformed from one of these forms to the other. We will also see that, in a closed system, the sum of these forms of energy remains constant.
Quite a bit of potential energy is gained by a roller coaster car and its passengers when they are raised to the top of the first hill. Remember that the potential part of the term means that energy has been stored and can be used at another time. You will see that this stored energy can either be used to do work or can be transformed into kinetic energy. For example, when an object that has gravitational potential energy falls, its energy is converted to kinetic energy. Remember that both work and energy are expressed in joules.
Refer back to . The amount of work required to raise the TV from point A to point B is equal to the amount of gravitational potential energy the TV gains from its height above the ground. This is generally true for any object raised above the ground. If all the work done on an object is used to raise the object above the ground, the amount work equals the object’s gain in gravitational potential energy. However, note that because of the work done by friction, these energy–work transformations are never perfect. Friction causes the loss of some useful energy. In the discussions to follow, we will use the approximation that transformations are frictionless.
Now, let’s look at the roller coaster in . Work was done on the roller coaster to get it to the top of the first rise; at this point, the roller coaster has gravitational potential energy. It is moving slowly, so it also has a small amount of kinetic energy. As the car descends the first slope, its PE is converted to KE. At the low point much of the original PE has been transformed to KE, and speed is at a maximum. As the car moves up the next slope, some of the KE is transformed back into PE and the car slows down.
On an actual roller coaster, there are many ups and downs, and each of these is accompanied by transitions between kinetic and potential energy. Assume that no energy is lost to friction. At any point in the ride, the total mechanical energy is the same, and it is equal to the energy the car had at the top of the first rise. This is a result of the law of conservation of energy, which says that, in a closed system, total energy is conservedthat is, it is constant. Using subscripts 1 and 2 to represent initial and final energy, this law is expressed as
Either side equals the total mechanical energy. The phrase in a means we are assuming no energy is lost to the surroundings due to friction and air resistance. If we are making calculations on dense falling objects, this is a good assumption. For the roller coaster, this assumption introduces some inaccuracy to the calculation.
### Calculations involving Mechanical Energy and Conservation of Energy
### Practice Problems
### Check Your Understanding
### Section Summary
1. Mechanical energy may be either kinetic (energy of motion) or potential (stored energy).
2. Doing work on an object or system changes its energy.
3. Total energy in a closed, isolated system is constant.
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# Work, Energy, and Simple Machines
## Simple Machines
### Section Key Terms
### Simple Machines
Simple machines make work easier, but they do not decrease the amount of work you have to do. Why can’t simple machines change the amount of work that you do? Recall that in closed systems the total amount of energy is conserved. A machine cannot increase the amount of energy you put into it. So, why is a simple machine useful? Although it cannot change the amount of work you do, a simple machine can change the amount of force you must apply to an object, and the distance over which you apply the force. In most cases, a simple machine is used to reduce the amount of force you must exert to do work. The down side is that you must exert the force over a greater distance, because the product of force and distance, fd, (which equals work) does not change.
Let’s examine how this works in practice. In (a), the worker uses a type of lever to exert a small force over a large distance, while the pry bar pulls up on the nail with a large force over a small distance. (b) shows the how a lever works mathematically. The effort force, applied at F, lifts the load (the resistance force) which is pushing down at F. The triangular pivot is called the fulcrum; the part of the lever between the fulcrum and F is the effort arm, L; and the part to the left is the resistance arm, L. The mechanical advantage is a number that tells us how many times a simple machine multiplies the effort force. The ideal mechanical advantage, IMA, is the mechanical advantage of a perfect machine with no loss of useful work caused by friction between moving parts. The equation for IMA is shown in (b).
In general, the IMA = the resistance force, F, divided by the effort force, F. IMA also equals the distance over which the effort is applied, d, divided by the distance the load travels, d.
Getting back to conservation of energy, for any simple machine, the work put into the machine, W, equals the work the machine puts out, W. Combining this with the information in the paragraphs above, we can write
The equations show how a simple machine can output the same amount of work while reducing the amount of effort force by increasing the distance over which the effort force is applied.
Some levers exert a large force to a short effort arm. This results in a smaller force acting over a greater distance at the end of the resistance arm. Examples of this type of lever are baseball bats, hammers, and golf clubs. In another type of lever, the fulcrum is at the end of the lever and the load is in the middle, as in the design of a wheelbarrow.
The simple machine shown in is called a wheel and axle. It is actually a form of lever. The difference is that the effort arm can rotate in a complete circle around the fulcrum, which is the center of the axle. Force applied to the outside of the wheel causes a greater force to be applied to the rope that is wrapped around the axle. As shown in the figure, the ideal mechanical advantage is calculated by dividing the radius of the wheel by the radius of the axle. Any crank-operated device is an example of a wheel and axle.
An inclined plane and a wedge are two forms of the same simple machine. A wedge is simply two inclined planes back to back. shows the simple formulas for calculating the IMAs of these machines. All sloping, paved surfaces for walking or driving are inclined planes. Knives and axe heads are examples of wedges.
The screw shown in is actually a lever attached to a circular inclined plane. Wood screws (of course) are also examples of screws. The lever part of these screws is a screw driver. In the formula for IMA, the distance between screw threads is called pitch and has the symbol P.
shows three different pulley systems. Of all simple machines, mechanical advantage is easiest to calculate for pulleys. Simply count the number of ropes supporting the load. That is the IMA. Once again we have to exert force over a longer distance to multiply force. To raise a load 1 meter with a pulley system you have to pull N meters of rope. Pulley systems are often used to raise flags and window blinds and are part of the mechanism of construction cranes.
A complex machine is a combination of two or more simple machines. The wire cutters in combine two levers and two wedges. Bicycles include wheel and axles, levers, screws, and pulleys. Cars and other vehicles are combinations of many machines.
### Calculating Mechanical Advantage and Efficiency of Simple Machines
In general, the IMA = the resistance force, Fr, divided by the effort force, Fe. IMA also equals the distance over which the effort is applied, d, divided by the distance the load travels, d.
Refer back to the discussions of each simple machine for the specific equations for the IMA for each type of machine.
No simple or complex machines have the actual mechanical advantages calculated by the IMA equations. In real life, some of the applied work always ends up as wasted heat due to friction between moving parts. Both the input work (W) and output work (W) are the result of a force, F, acting over a distance, d.
The efficiency output of a machine is simply the output work divided by the input work, and is usually multiplied by 100 so that it is expressed as a percent.
Look back at the pictures of the simple machines and think about which would have the highest efficiency. Efficiency is related to friction, and friction depends on the smoothness of surfaces and on the area of the surfaces in contact. How would lubrication affect the efficiency of a simple machine?
### Practice Problems
### Check Your Understanding
### Section Summary
1. The six types of simple machines make work easier by changing the fd term so that force is reduced at the expense of increased distance.
2. The ratio of output force to input force is a machine’s mechanical advantage.
3. Combinations of two or more simple machines are called complex machines.
4. The ratio of output work to input work is a machine’s efficiency.
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# Special Relativity
## Introduction
Have you ever dreamed of traveling to other planets in faraway star systems? The trip might seem possible by traveling fast enough, but you will read in this chapter why it is not. In 1905, Albert Einstein developed the theory of special relativity. Einstein developed the theory to help explain inconsistencies between the equations describing electromagnetism and Newtonian mechanics, and to explain why the ether did not exist. This theory explains the limit on an object’s speed among other implications.
Relativity is the study of how different observers moving with respect to one another measure the same events. Galileo and Newton developed the first correct version of classical relativity. Einstein developed the modern theory of relativity. Modern relativity is divided into two parts. Special relativity deals with observers moving at constant velocity. General relativity deals with observers moving at constant acceleration. Einstein’s theories of relativity made revolutionary predictions. Most importantly, his predictions have been verified by experiments.
In this chapter, you learn how experiments and puzzling contradictions in existing theories led to the development of the theory of special relativity. You will also learn the simple postulates on which the theory was based; a postulate is a statement that is assumed to be true for the purposes of reasoning in a scientific or mathematic argument. |
# Special Relativity
## Postulates of Special Relativity
### Section Key Terms
### Scientific Experiments and Problems
Relativity is not new. Way back around the year 1600, Galileo explained that motion is relative. Wherever you happen to be, it seems like you are at a fixed point and that everything moves with respect to you. Everyone else feels the same way. Motion is always measured with respect to a fixed point. This is called establishing a frame of reference. But the choice of the point is arbitrary, and all frames of reference are equally valid. A passenger in a moving car is not moving with respect to the driver, but they are both moving from the point of view of a person on the sidewalk waiting for a bus. They are moving even faster as seen by a person in a car coming toward them. It is all relative.
Light is involved in the discussion of relativity because theories related to electromagnetism are inconsistent with Galileo’s and Newton’s explanation of relativity. The true nature of light was a hot topic of discussion and controversy in the late 19th century. At the time, it was not generally believed that light could travel across empty space. It was known to travel as waves, and all other types of energy that propagated as waves needed to travel though a material medium. It was believed that space was filled with an invisible medium that light waves traveled through. This imaginary (as it turned out) material was called the ether (also spelled aether). It was thought that everything moved through this mysterious fluid. In other words, ether was the one fixed frame of reference. The Michelson–Morley experiment proved it was not.
In 1887, Albert Michelson and Edward Morley designed the interferometer shown in to measure the speed of Earth through the ether. A light beam is split into two perpendicular paths and then recombined. Recombining the waves produces an inference pattern, with a bright fringe at the locations where the two waves arrive in phase; that is, with the crests of both waves arriving together and the troughs arriving together. A dark fringe appears where the crest of one wave coincides with a trough of the other, so that the two cancel. If Earth is traveling through the ether as it orbits the sun, the peaks in one arm would take longer than in the other to reach the same location. The places where the two waves arrive in phase would change, and the interference pattern would shift. But, using the interferometer, there was no shift seen! This result led to two conclusions: that there is no ether and that the speed of light is the same regardless of the relative motion of source and observer. The Michelson–Morley investigation has been called the most famous failed experiment in history.
To see what Michelson and Morley expected to find when they measured the speed of light in two directions, watch this animation. In the video, two people swimming in a lake are represented as an analogy to light beams leaving Earth as it moves through the ether (if there were any ether). The swimmers swim away from and back to a platform that is moving through the water. The swimmers swim in different directions with respect to the motion of the platform. Even though they swim equal distances at the same speed, the motion of the platform causes them to arrive at different times.
### Einstein’s Postulates
The results described above left physicists with some puzzling and unsettling questions such as, why doesn’t light emitted by a fast-moving object travel faster than light from a street lamp? A radical new theory was needed, and Albert Einstein, shown in , was about to become everyone’s favorite genius. Einstein began with two simple postulates based on the two things we have discussed so far in this chapter.
1. The laws of physics are the same in all inertial reference frames.
2. The speed of light is the same in all inertial reference frames and is not affected by the speed of its source.
The speed of light is given the symbol c and is equal to exactly 299,792,458 m/s. This is the speed of light in vacuum; that is, in the absence of air. For most purposes, we round this number off to
The term inertial reference frame simply refers to a frame of reference where all objects follow Newton’s first law of motion: Objects at rest remain at rest, and objects in motion remain in motion at a constant velocity in a straight line, unless acted upon by an external force. The inside of a car moving along a road at constant velocity and the inside of a stationary house are inertial reference frames.
Einstein’s postulates were carefully chosen, and they both seemed very likely to be true. Einstein proceeded despite realizing that these two ideas taken together and applied to extreme conditions led to results that contradict Newtonian mechanics. He just took the ball and ran with it.
In the traditional view, velocities are additive. If you are running at 3 m/s and you throw a ball forward at a speed of 10 m/s, the ball should have a net speed of 13 m/s. However, according to relativity theory, the speed of a moving light source is not added to the speed of the emitted light.
In addition, Einstein’s theory shows that if you were moving forward relative to Earth at nearly c (the speed of light) and could throw a ball forward at c, an observer at rest on the earth would not see the ball moving at nearly twice the speed of light. The observer would see it moving at a speed that is still less than c. This result conforms to both of Einstein’s postulates: The speed of light has a fixed maximum and neither reference frame is privileged.
Consider how we measure elapsed time. If we use a stopwatch, for example, how do we know when to start and stop the watch? One method is to use the arrival of light from the event, such as observing a light turn green to start a drag race. The timing will be more accurate if some sort of electronic detection is used, avoiding human reaction times and other complications.
Now suppose we use this method to measure the time interval between two flashes of light produced by flash lamps on a moving train. (See )
A woman (observer A) is seated in the center of a rail car, with two flash lamps at opposite sides equidistant from her. Multiple light rays that are emitted from the flash lamps move towards observer A, as shown with arrows. A velocity vector arrow for the rail car is shown towards the right. A man (observer B) standing on the platform is facing the woman and also observes the flashes of light.
Observer A moves with the lamps on the rail car as the rail car moves towards the right of observer B. Observer B receives the light flashes simultaneously, and sees the bulbs as both having flashed at the same time. However, he sees observer A receive the flash from the right first. Because the pulse from the right reaches her first, in her frame of reference she sees the bulbs as not having flashed simultaneously. Here, a relative velocity between observers affects whether two events at well-separated locations are observed to be simultaneous. Simultaneity, or whether different events occur at the same instant, depends on the frame of reference of the observer. Remember that velocity equals distance divided by time, so t = d/v. If velocity appears to be different, then duration of time appears to be different.
This illustrates the power of clear thinking. We might have guessed incorrectly that, if light is emitted simultaneously, then two observers halfway between the sources would see the flashes simultaneously. But careful analysis shows this not to be the case. Einstein was brilliant at this type of thought experiment (in German, Gedankenexperiment). He very carefully considered how an observation is made and disregarded what might seem obvious. The validity of thought experiments, of course, is determined by actual observation. The genius of Einstein is evidenced by the fact that experiments have repeatedly confirmed his theory of relativity. No experiments after that of Michelson and Morley were able to detect any ether medium. We will describe later how experiments also confirmed other predictions of special relativity, such as the distance between two objects and the time interval of two events being different for two observers moving with respect to each other.
In summary: Two events are defined to be simultaneous if an observer measures them as occurring at the same time (such as by receiving light from the events). Two events are not necessarily simultaneous to all observers.
The discrepancies between Newtonian mechanics and relativity theory illustrate an important point about how science advances. Einstein’s theory did not replace Newton’s but rather extended it. It is not unusual that a new theory must be developed to account for new information. In most cases, the new theory is built on the foundation of older theory. It is rare that old theories are completely replaced.
In this chapter, you will learn about the theory of special relativity, but, as mentioned in the introduction, Einstein developed two relativity theories: special and general. summarizes the differences between the two theories.
### Practice Problems
### Check Your Understanding
### Section Summary
1. One postulate of special relativity theory is that the laws of physics are the same in all inertial frames of reference.
2. The other postulate is that the speed of light in a vacuum is the same in all inertial frames.
3. Einstein showed that simultaneity, or lack of it, depends on the frame of reference of the observer.
### Key Equations
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# Special Relativity
## Consequences of Special Relativity
### Section Key Terms
### Relativistic Effects on Time, Distance, and Momentum
Consideration of the measurement of elapsed time and simultaneity leads to an important relativistic effect. Time dilation is the phenomenon of time passing more slowly for an observer who is moving relative to another observer.
For example, suppose an astronaut measures the time it takes for light to travel from the light source, cross her ship, bounce off a mirror, and return. (See .) How does the elapsed time the astronaut measures compare with the elapsed time measured for the same event by a person on the earth? Asking this question (another thought experiment) produces a profound result. We find that the elapsed time for a process depends on who is measuring it. In this case, the time measured by the astronaut is smaller than the time measured by the earth bound observer. The passage of time is different for the two observers because the distance the light travels in the astronaut’s frame is smaller than in the earth bound frame. Light travels at the same speed in each frame, and so it will take longer to travel the greater distance in the earth bound frame.
The relationship between Δt and Δto is given by
where
is the relativistic factor given by
and v and c are the speeds of the moving observer and light, respectively.
Notice that when the velocity v is small compared to the speed of light c, then v/c becomes small, and
becomes close to 1. When this happens, time measurements are the same in both frames of reference. Relativistic effects, meaning those that have to do with special relativity, usually become significant when speeds become comparable to the speed of light. This is seen to be the case for time dilation.
You may have seen science fiction movies in which space travelers return to Earth after a long trip to find that the planet and everyone on it has aged much more than they have. This type of scenario is a based on a thought experiment, known as the twin paradox, which imagines a pair of twins, one of whom goes on a trip into space while the other stays home. When the space traveler returns, she finds her twin has aged much more than she. This happens because the traveling twin has been in two frames of reference, one leaving Earth and one returning.
Time dilation has been confirmed by comparing the time recorded by an atomic clock sent into orbit to the time recorded by a clock that remained on Earth. GPS satellites must also be adjusted to compensate for time dilation in order to give accurate positioning.
Have you ever driven on a road, like that shown in , that seems like it goes on forever? If you look ahead, you might say you have about 10 km left to go. Another traveler might say the road ahead looks like it is about 15 km long. If you both measured the road, however, you would agree. Traveling at everyday speeds, the distance you both measure would be the same. You will read in this section, however, that this is not true at relativistic speeds. Close to the speed of light, distances measured are not the same when measured by different observers moving with respect to one other.
One thing all observers agree upon is their relative speed. When one observer is traveling away from another, they both see the other receding at the same speed, regardless of whose frame of reference is chosen. Remember that speed equals distance divided by time: v = d/t. If the observers experience a difference in elapsed time, they must also observe a difference in distance traversed. This is because the ratio d/t must be the same for both observers.
The shortening of distance experienced by an observer moving with respect to the points whose distance apart is measured is called length contraction. Proper length, L0, is the distance between two points measured in the reference frame where the observer and the points are at rest. The observer in motion with respect to the points measures L. These two lengths are related by the equation
Because
is the same expression used in the time dilation equation above, the equation becomes
To see how length contraction is seen by a moving observer, go to this simulation. Here you can also see that simultaneity, time dilation, and length contraction are interrelated phenomena.
This link is to a simulation that illustrates the relativity of simultaneous events.
In classical physics, momentum is a simple product of mass and velocity. When special relativity is taken into account, objects that have mass have a speed limit. What effect do you think mass and velocity have on the momentum of objects moving at relativistic speeds; i.e., speeds close to the speed of light?
Momentum is one of the most important concepts in physics. The broadest form of Newton’s second law is stated in terms of momentum. Momentum is conserved in classical mechanics whenever the net external force on a system is zero. This makes momentum conservation a fundamental tool for analyzing collisions. We will see that momentum has the same importance in modern physics. Relativistic momentum is conserved, and much of what we know about subatomic structure comes from the analysis of collisions of accelerator-produced relativistic particles.
One of the postulates of special relativity states that the laws of physics are the same in all inertial frames. Does the law of conservation of momentum survive this requirement at high velocities? The answer is yes, provided that the momentum is defined as follows.
Relativistic momentum, p, is classical momentum multiplied by the relativistic factor
where
is the rest mass of the object (that is, the mass measured at rest, without any
factor involved),
is its velocity relative to an observer, and
as before, is the relativistic factor. We use the mass of the object as measured at rest because we cannot determine its mass while it is moving.
Note that we use
for velocity here to distinguish it from relative velocity
between observers. Only one observer is being considered here. With
defined in this way,
is conserved whenever the net external force is zero, just as in classical physics. Again we see that the relativistic quantity becomes virtually the same as the classical at low velocities. That is, relativistic momentum
becomes the classical
at low velocities, because
is very nearly equal to 1 at low velocities.
Relativistic momentum has the same intuitive feel as classical momentum. It is greatest for large masses moving at high velocities. Because of the factor
however, relativistic momentum behaves differently from classical momentum by approaching infinity as
approaches
(See .) This is another indication that an object with mass cannot reach the speed of light. If it did, its momentum would become infinite, which is an unreasonable value.
Relativistic momentum is defined in such a way that the conservation of momentum will hold in all inertial frames. Whenever the net external force on a system is zero, relativistic momentum is conserved, just as is the case for classical momentum. This has been verified in numerous experiments.
### Mass-Energy Equivalence
Let us summarize the calculation of relativistic effects on objects moving at speeds near the speed of light. In each case we will need to calculate the relativistic factor, given by
where v and c are as defined earlier. We use u as the velocity of a particle or an object in one frame of reference, and v for the velocity of one frame of reference with respect to another.
### Time Dilation
Elapsed time on a moving object,
as seen by a stationary observer is given by
where
is the time observed on the moving object when it is taken to be the frame or reference.
### Length Contraction
Length measured by a person at rest with respect to a moving object, L, is given by
where L0 is the length measured on the moving object.
### Relativistic Momentum
Momentum, p, of an object of mass, m, traveling at relativistic speeds is given by
where u is velocity of a moving object as seen by a stationary observer.
### Relativistic Energy
The original source of all the energy we use is the conversion of mass into energy. Most of this energy is generated by nuclear reactions in the sun and radiated to Earth in the form of electromagnetic radiation, where it is then transformed into all the forms with which we are familiar. The remaining energy from nuclear reactions is produced in nuclear power plants and in Earth’s interior. In each of these cases, the source of the energy is the conversion of a small amount of mass into a large amount of energy. These sources are shown in .
The first postulate of relativity states that the laws of physics are the same in all inertial frames. Einstein showed that the law of conservation of energy is valid relativistically, if we define energy to include a relativistic factor. The result of his analysis is that a particle or object of mass m moving at velocity u has relativistic energy given by
This is the expression for the total energy of an object of mass m at any speed u and includes both kinetic and potential energy. Look back at the equation for
and you will see that it is equal to 1 when u is 0; that is, when an object is at rest. Then the rest energy, E0, is simply
This is the correct form of Einstein’s famous equation.
This equation is very useful to nuclear physicists because it can be used to calculate the energy released by a nuclear reaction. This is done simply by subtracting the mass of the products of such a reaction from the mass of the reactants. The difference is the m in
Here is a simple example:
A positron is a type of antimatter that is just like an electron, except that it has a positive charge. When a positron and an electron collide, their masses are completely annihilated and converted to energy in the form of gamma rays. Because both particles have a rest mass of 9.11 × 10–31 kg, we multiply the mc2 term by 2. So the energy of the gamma rays is
where we have the expression for the joule (J) in terms of its SI base units of kg, m, and s. In general, the nuclei of stable isotopes have less mass then their constituent subatomic particles. The energy equivalent of this difference is called the binding energy of the nucleus. This energy is released during the formation of the isotope from its constituent particles because the product is more stable than the reactants. Expressed as mass, it is called the mass defect. For example, a helium nucleus is made of two neutrons and two protons and has a mass of 4.0003 atomic mass units (u). The sum of the masses of two protons and two neutrons is 4.0330 u. The mass defect then is 0.0327 u. Converted to kg, the mass defect is 5.0442 × 10–30 kg. Multiplying this mass times c2 gives a binding energy of 4.540 × 10–12 J. This does not sound like much because it is only one atom. If you were to make one gram of helium out of neutrons and protons, it would release 683,000,000,000 J. By comparison, burning one gram of coal releases about 24 J.
### Practice Problems
### Check Your Understanding
### Section Summary
1. Time dilates, length contracts, and momentum increases as an object approaches the speed of light.
2. Energy and mass are interchangeable, according to the relationship E = mc2. The laws of conservation of mass and energy are combined into the law of conservation of mass-energy.
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# Thermal Energy, Heat, and Work
## Introduction
Heat is something familiar to all of us. We feel the warmth of the summer sun, the hot vapor rising up out of a cup of hot cocoa, and the cooling effect of our sweat. When we feel warmth, it means that heat is transferring energy to our bodies; when we feel cold, that means heat is transferring energy away from our bodies. Heat transfer is the movement of thermal energy from one place or material to another, and is caused by temperature differences. For example, much of our weather is caused by Earth evening out the temperature across the planet through wind and violent storms, which are driven by heat transferring energy away from the equator towards the cold poles. In this chapter, we’ll explore the precise meaning of heat, how it relates to temperature as well as to other forms of energy, and its connection to work. |
# Thermal Energy, Heat, and Work
## Temperature and Thermal Energy
### Section Key Terms
### Temperature
What is temperature? It’s one of those concepts so ingrained in our everyday lives that, although we know what it means intuitively, it can be hard to define. It is tempting to say that temperature measures heat, but this is not strictly true. Heat is the transfer of energy due to a temperature difference. Temperature is defined in terms of the instrument we use to tell us how hot or cold an object is, based on a mechanism and scale invented by people. Temperature is literally defined as what we measure on a thermometer.
Heat is often confused with temperature. For example, we may say that the heat was unbearable, when we actually mean that the temperature was high. This is because we are sensitive to the flow of energy by heat, rather than the temperature. Since heat, like work, transfers energy, it has the SI unit of joule (J).
Atoms and molecules are constantly in motion, bouncing off one another in random directions. Recall that kinetic energy is the energy of motion, and that it increases in proportion to velocity squared. Without going into mathematical detail, we can say that thermal energy—the energy associated with heat—is the average kinetic energy of the particles (molecules or atoms) in a substance. Faster moving molecules have greater kinetic energies, and so the substance has greater thermal energy, and thus a higher temperature. The total internal energy of a system is the sum of the kinetic and potential energies of its atoms and molecules. Thermal energy is one of the subcategories of internal energy, as is chemical energy.
To measure temperature, some scale must be used as a standard of measurement. The three most commonly used temperature scales are the Fahrenheit, Celsius, and Kelvin scales. Both the Fahrenheit scale and Celsius scale are relative temperature scales, meaning that they are made around a reference point. For example, the Celsius scale uses the freezing point of water as its reference point; all measurements are either lower than the freezing point of water by a given number of degrees (and have a negative sign), or higher than the freezing point of water by a given number of degrees (and have a positive sign). The boiling point of water is 100
for the Celsius scale, and its unit is the degree Celsius .
On the Fahrenheit scale, the freezing point of water is at 32
, and the boiling point is at 212
. The unit of temperature on this scale is the degree Fahrenheit . Note that the difference in degrees between the freezing and boiling points is greater for the Fahrenheit scale than for the Celsius scale. Therefore, a temperature difference of one degree Celsius is greater than a temperature difference of one degree Fahrenheit. Since 100 Celsius degrees span the same range as 180 Fahrenheit degrees, one degree on the Celsius scale is 1.8 times larger than one degree on the Fahrenheit scale (because
). This relationship can be used to convert between temperatures in Fahrenheit and Celsius (see ).
The Kelvin scale is the temperature scale that is commonly used in science because it is an absolute temperature scale. This means that the theoretically lowest-possible temperature is assigned the value of zero. Zero degrees on the Kelvin scale is known as absolute zero; it is theoretically the point at which there is no molecular motion to produce thermal energy. On the original Kelvin scale first created by Lord Kelvin, all temperatures have positive values, making it useful for scientific work. The official temperature unit on this scale is the kelvin, which is abbreviated as K. The freezing point of water is 273.15 K, and the boiling point of water is 373.15 K.
Although absolute zero is possible in theory, it cannot be reached in practice. The lowest temperature ever created and measured during a laboratory experiment was
K, at Helsinki University of Technology in Finland. In comparison, the coldest recorded temperature for a place on Earth’s surface was 183 K (–89 °C
), at Vostok, Antarctica, and the coldest known place (outside the lab) in the universe is the Boomerang Nebula, with a temperature of 1 K. Luckily, most of us humans will never have to experience such extremes.
The average normal body temperature is 98.6
(37.0
), but people have been known to survive with body temperatures ranging from 75
to 111
(24
to 44
).
### Converting Between Celsius, Kelvin, and Fahrenheit Scales
While the Fahrenheit scale is still the most commonly used scale in the United States, the majority of the world uses Celsius, and scientists prefer Kelvin. It’s often necessary to convert between these scales. For instance, if the TV meteorologist gave the local weather report in kelvins, there would likely be some confused viewers! gives the equations for conversion between the three temperature scales.
### Practice Problems
### Check Your Understanding
### Section Summary
1. Temperature is the quantity measured by a thermometer.
2. Temperature is related to the average kinetic energy of atoms and molecules in a system.
3. Absolute zero is the temperature at which there is no molecular motion.
4. There are three main temperature scales: Celsius, Fahrenheit, and Kelvin.
5. Temperatures on one scale can be converted into temperatures on another scale.
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# Thermal Energy, Heat, and Work
## Heat, Specific Heat, and Heat Transfer
### Section Key Terms
### Heat Transfer, Specific Heat, and Heat Capacity
We learned in the previous section that temperature is proportional to the average kinetic energy of atoms and molecules in a substance, and that the average internal kinetic energy of a substance is higher when the substance’s temperature is higher.
If two objects at different temperatures are brought in contact with each other, energy is transferred from the hotter object (that is, the object with the greater temperature) to the colder (lower temperature) object, until both objects are at the same temperature. There is no net heat transfer once the temperatures are equal because the amount of heat transferred from one object to the other is the same as the amount of heat returned. One of the major effects of heat transfer is temperature change: Heating increases the temperature while cooling decreases it. Experiments show that the heat transferred to or from a substance depends on three factors—the change in the substance’s temperature, the mass of the substance, and certain physical properties related to the phase of the substance.
The equation for heat transfer Q is
where m is the mass of the substance and ΔT is the change in its temperature, in units of Celsius or Kelvin. The symbol c stands for specific heat, and depends on the material and phase. The specific heat is the amount of heat necessary to change the temperature of 1.00 kg of mass by 1.00 ºC. The specific heat c is a property of the substance; its SI unit is J/(kg
K) or J/(kg
). The temperature change (
) is the same in units of kelvins and degrees Celsius (but not degrees Fahrenheit). Specific heat is closely related to the concept of heat capacity. Heat capacity is the amount of heat necessary to change the temperature of a substance by 1.00
. In equation form, heat capacity C is
, where m is mass and c is specific heat. Note that heat capacity is the same as specific heat, but without any dependence on mass. Consequently, two objects made up of the same material but with different masses will have different heat capacities. This is because the heat capacity is a property of an object, but specific heat is a property of any object made of the same material.
Values of specific heat must be looked up in tables, because there is no simple way to calculate them. gives the values of specific heat for a few substances as a handy reference. We see from this table that the specific heat of water is five times that of glass, which means that it takes five times as much heat to raise the temperature of 1 kg of water than to raise the temperature of 1 kg of glass by the same number of degrees.
### Conduction, Convection, and Radiation
Whenever there is a temperature difference, heat transfer occurs. Heat transfer may happen rapidly, such as through a cooking pan, or slowly, such as through the walls of an insulated cooler.
There are three different heat transfer methods: conduction, convection, and radiation. At times, all three may happen simultaneously. See .
Conduction is heat transfer through direct physical contact. Heat transferred between the electric burner of a stove and the bottom of a pan is transferred by conduction. Sometimes, we try to control the conduction of heat to make ourselves more comfortable. Since the rate of heat transfer is different for different materials, we choose fabrics, such as a thick wool sweater, that slow down the transfer of heat away from our bodies in winter.
As you walk barefoot across the living room carpet, your feet feel relatively comfortable…until you step onto the kitchen’s tile floor. Since the carpet and tile floor are both at the same temperature, why does one feel colder than the other? This is explained by different rates of heat transfer: The tile material removes heat from your skin at a greater rate than the carpeting, which makes it feel colder.
Some materials simply conduct thermal energy faster than others. In general, metals (like copper, aluminum, gold, and silver) are good heat conductors, whereas materials like wood, plastic, and rubber are poor heat conductors.
shows particles (either atoms or molecules) in two bodies at different temperatures. The (average) kinetic energy of a particle in the hot body is higher than in the colder body. If two particles collide, energy transfers from the particle with greater kinetic energy to the particle with less kinetic energy. When two bodies are in contact, many particle collisions occur, resulting in a net flux of heat from the higher-temperature body to the lower-temperature body. The heat flux depends on the temperature difference
. Therefore, you will get a more severe burn from boiling water than from hot tap water.
Convection is heat transfer by the movement of a fluid. This type of heat transfer happens, for example, in a pot boiling on the stove, or in thunderstorms, where hot air rises up to the base of the clouds.
As the temperature of fluids increase, they expand and become less dense. For example, could represent the wall of a balloon with different temperature gases inside the balloon than outside in the environment. The hotter and thus faster moving gas particles inside the balloon strike the surface with more force than the cooler air outside, causing the balloon to expand. This decrease in density relative to its environment creates buoyancy (the tendency to rise). Convection is driven by buoyancy—hot air rises because it is less dense than the surrounding air.
Sometimes, we control the temperature of our homes or ourselves by controlling air movement. Sealing leaks around doors with weather stripping keeps out the cold wind in winter. The house in and the pot of water on the stove in are both examples of convection and buoyancy by human design. Ocean currents and large-scale atmospheric circulation transfer energy from one part of the globe to another, and are examples of natural convection.
Radiation is a form of heat transfer that occurs when electromagnetic radiation is emitted or absorbed. Electromagnetic radiation includes radio waves, microwaves, infrared radiation, visible light, ultraviolet radiation, X-rays, and gamma rays, all of which have different wavelengths and amounts of energy (shorter wavelengths have higher frequency and more energy).
You can feel the heat transfer from a fire and from the sun. Similarly, you can sometimes tell that the oven is hot without touching its door or looking inside—it may just warm you as you walk by. Another example is thermal radiation from the human body; people are constantly emitting infrared radiation, which is not visible to the human eye, but is felt as heat.
Radiation is the only method of heat transfer where no medium is required, meaning that the heat doesn’t need to come into direct contact with or be transported by any matter. The space between Earth and the sun is largely empty, without any possibility of heat transfer by convection or conduction. Instead, heat is transferred by radiation, and Earth is warmed as it absorbs electromagnetic radiation emitted by the sun.
All objects absorb and emit electromagnetic radiation (see ). The rate of heat transfer by radiation depends mainly on the color of the object. Black is the most effective absorber and radiator, and white is the least effective. People living in hot climates generally avoid wearing black clothing, for instance. Similarly, black asphalt in a parking lot will be hotter than adjacent patches of grass on a summer day, because black absorbs better than green. The reverse is also true—black radiates better than green. On a clear summer night, the black asphalt will be colder than the green patch of grass, because black radiates energy faster than green. In contrast, white is a poor absorber and also a poor radiator. A white object reflects nearly all radiation, like a mirror.
### Solving Heat Transfer Problems
### Practice Problems
### Check Your Understanding
### Section Summary
1. Heat is thermal (internal) energy transferred due to a temperature difference.
2. The transfer of heat Q that leads to a change
in the temperature of a body with mass m is
, where c is the specific heat of the material.
3. Heat is transferred by three different methods: conduction, convection, and radiation.
4. Heat conduction is the transfer of heat between two objects in direct contact with each other.
5. Convection is heat transfer by the movement of mass.
6. Radiation is heat transfer by electromagnetic waves.
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# Thermal Energy, Heat, and Work
## Phase Change and Latent Heat
### Section Key Terms
### Phase Changes
So far, we have learned that adding thermal energy by heat increases the temperature of a substance. But surprisingly, there are situations where adding energy does not change the temperature of a substance at all! Instead, the additional thermal energy acts to loosen bonds between molecules or atoms and causes a phase change. Because this energy enters or leaves a system during a phase change without causing a temperature change in the system, it is known as latent heat (latent means hidden).
The three phases of matter that you frequently encounter are solid, liquid and gas (see ). Solid has the least energetic state; atoms in solids are in close contact, with forces between them that allow the particles to vibrate but not change position with neighboring particles. (These forces can be thought of as springs that can be stretched or compressed, but not easily broken.)
Liquid has a more energetic state, in which particles can slide smoothly past one another and change neighbors, although they are still held together by their mutual attraction.
Gas has a more energetic state than liquid, in which particles are broken free of their bonds. Particles in gases are separated by distances that are large compared with the size of the particles.
The most energetic state of all is plasma. Although you may not have heard much about plasma, it is actually the most common state of matter in the universe—stars are made up of plasma, as is lightning. The plasma state is reached by heating a gas to the point where particles are pulled apart, separating the electrons from the rest of the particle. This produces an ionized gas that is a combination of the negatively charged free electrons and positively charged ions, known as plasma.
During a phase change, matter changes from one phase to another, either through the addition of energy by heat and the transition to a more energetic state, or from the removal of energy by heat and the transition to a less energetic state.
Phase changes to a more energetic state include the following:
1. Melting—Solid to liquid
2. Vaporization—Liquid to gas (included boiling and evaporation)
3. Sublimation—Solid to gas
4. IonizationGas to plasma
Phase changes to a less energetic state are as follows:
1. Condensation—Gas to liquid
2. Freezing—Liquid to solid
3. Recombination—Plasma to gas
4. DepositionGas to solid
Energy is required to melt a solid because the bonds between the particles in the solid must be broken. Since the energy involved in a phase changes is used to break bonds, there is no increase in the kinetic energies of the particles, and therefore no rise in temperature. Similarly, energy is needed to vaporize a liquid to overcome the attractive forces between particles in the liquid. There is no temperature change until a phase change is completed. The temperature of a cup of soda and ice that is initially at 0
stays at 0
until all of the ice has melted. In the reverse of these processes—freezing and condensation—energy is released from the latent heat (see ).
The heat, Q, required to change the phase of a sample of mass m is
(for melting/freezing),
(for vaporization/condensation),
where
is the latent heat of fusion, and
is the latent heat of vaporization. The latent heat of fusion is the amount of heat needed to cause a phase change between solid and liquid. The latent heat of vaporization is the amount of heat needed to cause a phase change between liquid and gas.
and
are coefficients that vary from substance to substance, depending on the strength of intermolecular forces, and both have standard units of J/kg. See for values of
and
of different substances.
Let’s consider the example of adding heat to ice to examine its transitions through all three phases—solid to liquid to gas. A phase diagram indicating the temperature changes of water as energy is added is shown in . The ice starts out at −20
, and its temperature rises linearly, absorbing heat at a constant rate until it reaches 0
Once at this temperature, the ice gradually melts, absorbing 334 kJ/kg. The temperature remains constant at 0
during this phase change. Once all the ice has melted, the temperature of the liquid water rises, absorbing heat at a new constant rate. At 100
, the water begins to boil and the temperature again remains constant while the water absorbs 2256 kJ/kg during this phase change. When all the liquid has become steam, the temperature rises again at a constant rate.
We have seen that vaporization requires heat transfer to a substance from its surroundings. Condensation is the reverse process, where heat in transferred away from a substance to its surroundings. This release of latent heat increases the temperature of the surroundings. Energy must be removed from the condensing particles to make a vapor condense. This is why condensation occurs on cold surfaces: the heat transfers energy away from the warm vapor to the cold surface. The energy is exactly the same as that required to cause the phase change in the other direction, from liquid to vapor, and so it can be calculated from
. Latent heat is also released into the environment when a liquid freezes, and can be calculated from
.
### Solving Thermal Energy Problems with Phase Changes
### Practice Problems
### Check Your Understanding
### Section Summary
1. Most substances have four distinct phases: solid, liquid, gas, and plasma.
2. Gas is the most energetic state and solid is the least.
3. During a phase change, a substance undergoes transition to a higher energy state when heat is added, or to a lower energy state when heat is removed.
4. Heat is added to a substance during melting and vaporization.
5. Latent heat is released by a substance during condensation and freezing.
6. Phase changes occur at fixed temperatures called boiling and freezing (or melting) points for a given substance.
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# Thermodynamics
## Introduction
Energy can be transferred to or from a system, either through a temperature difference between it and another system (i.e., by heat) or by exerting a force through a distance (work). In these ways, energy can be converted into other forms of energy in other systems. For example, a car engine burns fuel for heat transfer into a gas. Work is done by the gas as it exerts a force through a distance by pushing a piston outward. This work converts the energy into a variety of other forms—into an increase in the car’s kinetic or gravitational potential energy; into electrical energy to run the spark plugs, radio, and lights; and back into stored energy in the car’s battery. But most of the thermal energy transferred by heat from the fuel burning in the engine does not do work on the gas. Instead, much of this energy is released into the surroundings at lower temperature (i.e., lost through heat), which is quite inefficient. Car engines are only about 25 to 30 percent efficient. This inefficiency leads to increased fuel costs, so there is great interest in improving fuel efficiency. However, it is common knowledge that modern gasoline engines cannot be made much more efficient. The same is true about the conversion to electrical energy in large power stations, whether they are coal, oil, natural gas, or nuclear powered. Why is this the case?
The answer lies in the nature of heat. Basic physical laws govern how heat transfer for doing work takes place and limit the maximum possible efficiency of the process. This chapter will explore these laws as well their applications to everyday machines. These topics are part of thermodynamics—the study of heat and its relationship to doing work. |
# Thermodynamics
## Zeroth Law of Thermodynamics: Thermal Equilibrium
### Section Key Terms
We learned in the previous chapter that when two objects (or systems) are in contact with one another, heat will transfer thermal energy from the object at higher temperature to the one at lower temperature until they both reach the same temperature. The objects are then in thermal equilibrium, and no further temperature changes will occur if they are isolated from other systems. The systems interact and change because their temperatures are different, and the changes stop once their temperatures are the same. Thermal equilibrium is established when two bodies are in thermal contact with each other—meaning heat transfer (i.e., the transfer of energy by heat) can occur between them. If two systems cannot freely exchange energy, they will not reach thermal equilibrium. (It is fortunate that empty space stands between Earth and the sun, because a state of thermal equilibrium with the sun would be too toasty for life on this planet!)
If two systems, A and B, are in thermal equilibrium with each other, and B is in thermal equilibrium with a third system, C, then A is also in thermal equilibrium with C. This statement may seem obvious, because all three have the same temperature, but it is basic to thermodynamics. It is called the zeroth law of thermodynamics.
You may be wondering at this point, why the wacky name? Shouldn’t this be called the first law of thermodynamics rather than the zeroth? The explanation is that this law was discovered after the first and second laws of thermodynamics but is so fundamental that scientists decided it should logically come first.
As an example of the zeroth law in action, consider newborn babies in neonatal intensive-care units in hospitals. Prematurely born or sick newborns are placed in special incubators. These babies have very little covering while in the incubators, so to an observer, they look as though they may not be warm enough. However, inside the incubator, the temperature of the air, the cot, and the baby are all the same—that is, they are in thermal equilibrium. The ambient temperature is just high enough to keep the baby safe and comfortable.
### Check Your Understanding
### Section Summary
1. Systems are in thermal equilibrium when they have the same temperature.
2. Thermal equilibrium occurs when two bodies are in contact with each other and can freely exchange energy.
3. The zeroth law of thermodynamics states that when two systems, A and B, are in thermal equilibrium with each other, and B is in thermal equilibrium with a third system, C, then A is also in thermal equilibrium with C.
### Concept Items
### Critical Thinking
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# Thermodynamics
## First law of Thermodynamics: Thermal Energy and Work
### Section Key Terms
### Pressure, Volume, Temperature, and the Ideal Gas Law
Before covering the first law of thermodynamics, it is first important to understand the relationship between pressure, volume, and temperature. Pressure, P, is defined as
where F is a force applied to an area, A, that is perpendicular to the force.
Depending on the area over which it is exerted, a given force can have a significantly different effect, as shown in .
The SI unit for pressure is the pascal, where
Pressure is defined for all states of matter but is particularly important when discussing fluids (such as air). You have probably heard the word pressure being used in relation to blood (high or low blood pressure) and in relation to the weather (high- and low-pressure weather systems). These are only two of many examples of pressures in fluids.
The relationship between the pressure, volume, and temperature for an ideal gas is given by the ideal gas law. A gas is considered ideal at low pressure and fairly high temperature, and forces between its component particles can be ignored. The ideal gas law states that
where P is the pressure of a gas, V is the volume it occupies, N is the number of particles (atoms or molecules) in the gas, and T is its absolute temperature. The constant k is called the Boltzmann constant and has the value
For the purposes of this chapter, we will not go into calculations using the ideal gas law. Instead, it is important for us to notice from the equation that the following are true for a given mass of gas:
1. When volume is constant, pressure is directly proportional to temperature.
2. When temperature is constant, pressure is inversely proportional to volume.
3. When pressure is constant, volume is directly proportional to temperature.
This last point describes thermal expansion—the change in size or volume of a given mass with temperature. What is the underlying cause of thermal expansion? An increase in temperature means that there’s an increase in the kinetic energy of the individual atoms. Gases are especially affected by thermal expansion, although liquids expand to a lesser extent with similar increases in temperature, and even solids have minor expansions at higher temperatures. This is why railroad tracks and bridges have expansion joints that allow them to freely expand and contract with temperature changes.
To get some idea of how pressure, temperature, and volume of a gas are related to one another, consider what happens when you pump air into a deflated tire. The tire’s volume first increases in direct proportion to the amount of air injected, without much increase in the tire pressure. Once the tire has expanded to nearly its full size, the walls limit volume expansion. If you continue to pump air into tire (which now has a nearly constant volume), the pressure increases with increasing temperature (see ).
### Pressure–Volume Work
Pressure–volume work is the work that is done by the compression or expansion of a fluid. Whenever there is a change in volume and external pressure remains constant, pressure–volume work is taking place. During a compression, a decrease in volume increases the internal pressure of a system as work is done on the system. During an expansion (), an increase in volume decreases the internal pressure of a system as the system does work.
Recall that the formula for work is
We can rearrange the definition of pressure,
to get an expression for force in terms of pressure.
Substituting this expression for force into the definition of work, we get
Because area multiplied by displacement is the change in volume,
, the mathematical expression for pressure–volume work is
Just as we say that work is force acting over a distance, for fluids, we can say that work is the pressure acting through the change in volume. For pressure–volume work, pressure is analogous to force, and volume is analogous to distance in the traditional definition of work.
### The First Law of Thermodynamics
Heat (Q) and work (W) are the two ways to add or remove energy from a system. The processes are very different. Heat is driven by temperature differences, while work involves a force exerted through a distance. Nevertheless, heat and work can produce identical results. For example, both can cause a temperature increase. Heat transfers energy into a system, such as when the sun warms the air in a bicycle tire and increases the air’s temperature. Similarly, work can be done on the system, as when the bicyclist pumps air into the tire. Once the temperature increase has occurred, it is impossible to tell whether it was caused by heat or work. Heat and work are both energy in transit—neither is stored as such in a system. However, both can change the internal energy, U, of a system.
Internal energy is the sum of the kinetic and potential energies of a system’s atoms and molecules. It can be divided into many subcategories, such as thermal and chemical energy, and depends only on the state of a system (that is, P, V, and T), not on how the energy enters or leaves the system.
In order to understand the relationship between heat, work, and internal energy, we use the first law of thermodynamics. The first law of thermodynamics applies the conservation of energy principle to systems where heat and work are the methods of transferring energy into and out of the systems. It can also be used to describe how energy transferred by heat is converted and transferred again by work.
The first law of thermodynamics states that the change in internal energy of a closed system equals the net heat transfer into the system minus the net work done by the system. In equation form, the first law of thermodynamics is
Here,
is the change in internal energy, U, of the system. As shown in , Q is the net heat transferred into the system—that is, Q is the sum of all heat transfers into and out of the system. W is the net work done by the system—that is, W is the sum of all work done on or by the system. By convention, if Q is positive, then there is a net heat transfer into the system; if W is positive, then there is net work done by the system. So positive Q adds energy to the system by heat, and positive W takes energy from the system by work. Note that if heat transfers more energy into the system than that which is done by work, the difference is stored as internal energy.
It follows also that negative Q indicates that energy is transferred away from the system by heat and so decreases the system’s internal energy, whereas negative W is work done on the system, which increases the internal energy.
### Solving Problems Involving the First Law of Thermodynamics
### Practice Problems
### Check Your Understanding
### Section Summary
1. Pressure is the force per unit area over which the force is applied perpendicular to the area.
2. Thermal expansion is the increase, or decrease, of the size (length, area, or volume) of a body due to a change in temperature.
3. The ideal gas law relates the pressure and volume of a gas to the number of gas particles (atoms or molecules) and the absolute temperature of the gas.
4. Heat and work are the two distinct methods of energy transfer.
5. Heat is energy transferred solely due to a temperature difference.
6. The first law of thermodynamics is given as
, where
is the change in internal energy of a system, Q is the net energy transfer into the system by heat (the sum of all transfers by heat into and out of the system), and W is the net work done by the system (the sum of all energy transfers by work out of or into the system).
7. Both Q and W represent energy in transit; only
represents an independent quantity of energy capable of being stored.
8. The internal energy U of a system depends only on the state of the system, and not how it reached that state.
### Key Equations
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# Thermodynamics
## Second Law of Thermodynamics: Entropy
### Section Key Terms
### Entropy
Recall from the chapter introduction that it is not even theoretically possible for engines to be 100 percent efficient. This phenomenon is explained by the second law of thermodynamics, which relies on a concept known as entropy. Entropy is a measure of the disorder of a system. Entropy also describes how much energy is not available to do work. The more disordered a system and higher the entropy, the less of a system's energy is available to do work.
Although all forms of energy can be used to do work, it is not possible to use the entire available energy for work. Consequently, not all energy transferred by heat can be converted into work, and some of it is lost in the form of waste heat—that is, heat that does not go toward doing work. The unavailability of energy is important in thermodynamics; in fact, the field originated from efforts to convert heat to work, as is done by engines.
The equation for the change in entropy,
, is
where Q is the heat that transfers energy during a process, and T is the absolute temperature at which the process takes place.
Q is positive for energy transferred into the system by heat and negative for energy transferred out of the system by heat. In SI, entropy is expressed in units of joules per kelvin (J/K). If temperature changes during the process, then it is usually a good approximation (for small changes in temperature) to take T to be the average temperature in order to avoid trickier math (calculus).
### Second Law of Thermodynamics
Have you ever played the card game 52 pickup? If so, you have been on the receiving end of a practical joke and, in the process, learned a valuable lesson about the nature of the universe as described by the second law of thermodynamics. In the game of 52 pickup, the prankster tosses an entire deck of playing cards onto the floor, and you get to pick them up. In the process of picking up the cards, you may have noticed that the amount of work required to restore the cards to an orderly state in the deck is much greater than the amount of work required to toss the cards and create the disorder.
The second law of thermodynamics states that the total entropy of a system either increases or remains constant in any spontaneous process; it never decreases. An important implication of this law is that heat transfers energy spontaneously from higher- to lower-temperature objects, but never spontaneously in the reverse direction. This is because entropy increases for heat transfer of energy from hot to cold (). Because the change in entropy is Q/T, there is a larger change in
at lower temperatures (smaller T). The decrease in entropy of the hot (larger T) object is therefore less than the increase in entropy of the cold (smaller T) object, producing an overall increase in entropy for the system.
Another way of thinking about this is that it is impossible for any process to have, as its sole result, heat transferring energy from a cooler to a hotter object. Heat cannot transfer energy spontaneously from colder to hotter, because the entropy of the overall system would decrease.
Suppose we mix equal masses of water that are originally at two different temperatures, say
and
. The result will be water at an intermediate temperature of
. Three outcomes have resulted: entropy has increased, some energy has become unavailable to do work, and the system has become less orderly. Let us think about each of these results.
First, why has entropy increased? Mixing the two bodies of water has the same effect as the heat transfer of energy from the higher-temperature substance to the lower-temperature substance. The mixing decreases the entropy of the hotter water but increases the entropy of the colder water by a greater amount, producing an overall increase in entropy.
Second, once the two masses of water are mixed, there is no more temperature difference left to drive energy transfer by heat and therefore to do work. The energy is still in the water, but it is now unavailable to do work.
Third, the mixture is less orderly, or to use another term, less structured. Rather than having two masses at different temperatures and with different distributions of molecular speeds, we now have a single mass with a broad distribution of molecular speeds, the average of which yields an intermediate temperature.
These three results—entropy, unavailability of energy, and disorder—not only are related but are, in fact, essentially equivalent. Heat transfer of energy from hot to cold is related to the tendency in nature for systems to become disordered and for less energy to be available for use as work.
Based on this law, what cannot happen? A cold object in contact with a hot one never spontaneously transfers energy by heat to the hot object, getting colder while the hot object gets hotter. Nor does a hot, stationary automobile ever spontaneously cool off and start moving.
Another example is the expansion of a puff of gas introduced into one corner of a vacuum chamber. The gas expands to fill the chamber, but it never regroups on its own in the corner. The random motion of the gas molecules could take them all back to the corner, but this is never observed to happen ().
We've explained that heat never transfers energy spontaneously from a colder to a hotter object. The key word here is spontaneously. If we do work on a system, it is possible to transfer energy by heat from a colder to hotter object. We'll learn more about this in the next section, covering refrigerators as one of the applications of the laws of thermodynamics.
Sometimes people misunderstand the second law of thermodynamics, thinking that based on this law, it is impossible for entropy to decrease at any particular location. But, it actually is possible for the entropy of one part of the universe to decrease, as long as the total change in entropy of the universe increases. In equation form, we can write this as
Based on this equation, we see that
can be negative as long as
is positive and greater in magnitude.
How is it possible for the entropy of a system to decrease? Energy transfer is necessary. If you pick up marbles that are scattered about the room and put them into a cup, your work has decreased the entropy of that system. If you gather iron ore from the ground and convert it into steel and build a bridge, your work has decreased the entropy of that system. Energy coming from the sun can decrease the entropy of local systems on Earth—that is,
is negative. But the overall entropy of the rest of the universe increases by a greater amount—that is,
is positive and greater in magnitude. In the case of the iron ore, although you made the system of the bridge and steel more structured, you did so at the expense of the universe. Altogether, the entropy of the universe is increased by the disorder created by digging up the ore and converting it to steel. Therefore,
and the second law of thermodynamics is not violated.
Every time a plant stores some solar energy in the form of chemical potential energy, or an updraft of warm air lifts a soaring bird, Earth experiences local decreases in entropy as it uses part of the energy transfer from the sun into deep space to do work. There is a large total increase in entropy resulting from this massive energy transfer. A small part of this energy transfer by heat is stored in structured systems on Earth, resulting in much smaller, local decreases in entropy.
### Solving Problems Involving the Second Law of Thermodynamics
Entropy is related not only to the unavailability of energy to do work; it is also a measure of disorder. For example, in the case of a melting block of ice, a highly structured and orderly system of water molecules changes into a disorderly liquid, in which molecules have no fixed positions (). There is a large increase in entropy for this process, as we'll see in the following worked example.
### Practice Problems
### Check Your Understanding
### Section Summary
1. Entropy is a measure of a system's disorder: the greater the disorder, the larger the entropy.
2. Entropy is also the reduced availability of energy to do work.
3. The second law of thermodynamics states that, for any spontaneous process, the total entropy of a system either increases or remains constant; it never decreases.
4. Heat transfers energy spontaneously from higher- to lower-temperature bodies, but never spontaneously in the reverse direction.
### Key Equations
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# Thermodynamics
## Applications of Thermodynamics: Heat Engines, Heat Pumps, and Refrigerators
### Section Key Terms
### Heat Engines, Heat Pumps, and Refrigerators
In this section, we’ll explore how heat engines, heat pumps, and refrigerators operate in terms of the laws of thermodynamics.
One of the most important things we can do with heat is to use it to do work for us. A heat engine does exactly this—it makes use of the properties of thermodynamics to transform heat into work. Gasoline and diesel engines, jet engines, and steam turbines that generate electricity are all examples of heat engines.
illustrates one of the ways in which heat transfers energy to do work. Fuel combustion releases chemical energy that heat transfers throughout the gas in a cylinder. This increases the gas temperature, which in turn increases the pressure of the gas and, therefore, the force it exerts on a movable piston. The gas does work on the outside world, as this force moves the piston through some distance. Thus, heat transfer of energy to the gas in the cylinder results in work being done.
To repeat this process, the piston needs to be returned to its starting point. Heat now transfers energy from the gas to the surroundings, so that the gas’s pressure decreases, and a force is exerted by the surroundings to push the piston back through some distance.
A cyclical process brings a system, such as the gas in a cylinder, back to its original state at the end of every cycle. All heat engines use cyclical processes.
Heat engines do work by using part of the energy transferred by heat from some source. As shown in , heat transfers energy,
, from the high-temperature object (or hot reservoir), whereas heat transfers unused energy,
, into the low-temperature object (or cold reservoir), and the work done by the engine is W. In physics, a reservoir is defined as an infinitely large mass that can take in or put out an unlimited amount of heat, depending upon the needs of the system. The temperature of the hot reservoir is
and the temperature of the cold reservoir is
.
As noted, a cyclical process brings the system back to its original condition at the end of every cycle. Such a system’s internal energy, U, is the same at the beginning and end of every cycle—that is,
. The first law of thermodynamics states that
where Q is the net heat transfer during the cycle, and W is the net work done by the system. The net heat transfer is the energy transferred in by heat from the hot reservoir minus the amount that is transferred out to the cold reservoir (
). Because there is no change in internal energy for a complete cycle (
), we have
so that
Therefore, the net work done by the system equals the net heat into the system, or
for a cyclical process.
Because the hot reservoir is heated externally, which is an energy-intensive process, it is important that the work be done as efficiently as possible. In fact, we want W to equal
, and for there to be no heat to the environment (that is,
). Unfortunately, this is impossible. According to the second law of thermodynamics, heat engines cannot have perfect conversion of heat into work. Recall that entropy is a measure of the disorder of a system, which is also how much energy is unavailable to do work. The second law of thermodynamics requires that the total entropy of a system either increases or remains constant in any process. Therefore, there is a minimum amount of
that cannot be used for work. The amount of heat rejected to the cold reservoir,
depends upon the efficiency of the heat engine. The smaller the increase in entropy,
, the smaller the value of
, and the more heat energy is available to do work.
Heat pumps, air conditioners, and refrigerators utilize heat transfer of energy from low to high temperatures, which is the opposite of what heat engines do. Heat transfers energy
from a cold reservoir and delivers energy
into a hot one. This requires work input, W, which produces a transfer of energy by heat. Therefore, the total heat transfer to the hot reservoir is
The purpose of a heat pump is to transfer energy by heat to a warm environment, such as a home in the winter. The great advantage of using a heat pump to keep your home warm rather than just burning fuel in a fireplace or furnace is that a heat pump supplies
. Heat
comes from the outside air, even at a temperature below freezing, to the indoor space. You only pay for W, and you get an additional heat transfer of
from the outside at no cost. In many cases, at least twice as much energy is transferred to the heated space as is used to run the heat pump. When you burn fuel to keep warm, you pay for all of it. The disadvantage to a heat pump is that the work input (required by the second law of thermodynamics) is sometimes more expensive than simply burning fuel, especially if the work is provided by electrical energy.
The basic components of a heat pump are shown in . A working fluid, such as a refrigerant, is used. In the outdoor coils (the evaporator), heat
enters the working fluid from the cold outdoor air, turning it into a gas.
The electrically driven compressor (work input W) raises the temperature and pressure of the gas and forces it into the condenser coils that are inside the heated space. Because the temperature of the gas is higher than the temperature inside the room, heat transfers energy to the room, and the gas condenses into a liquid. The liquid then flows back through an expansion (pressure-reducing) valve. The liquid, having been cooled through expansion, returns to the outdoor evaporator coils to resume the cycle.
The quality of a heat pump is judged by how much energy is transferred by heat into the warm space (
) compared with how much input work (W) is required.
Air conditioners and refrigerators are designed to cool substances by transferring energy by heat
out of a cool environment to a warmer one, where heat
is given up. In the case of a refrigerator, heat is moved out of the inside of the fridge into the surrounding room. For an air conditioner, heat is transferred outdoors from inside a home. Heat pumps are also often used in a reverse setting to cool rooms in the summer.
As with heat pumps, work input is required for heat transfer of energy from cold to hot. The quality of air conditioners and refrigerators is judged by how much energy is removed by heat
from a cold environment, compared with how much work, W, is required. So, what is considered the energy benefit in a heat pump, is considered waste heat in a refrigerator.
### Thermal Efficiency
In the conversion of energy into work, we are always faced with the problem of getting less out than we put in. The problem is that, in all processes, there is some heat
that transfers energy to the environment—and usually a very significant amount at that. A way to quantify how efficiently a machine runs is through a quantity called thermal efficiency.
We define thermal efficiency, Eff, to be the ratio of useful energy output to the energy input (or, in other words, the ratio of what we get to what we spend). The efficiency of a heat engine is the output of net work, W, divided by heat-transferred energy,
, into the engine; that is
An efficiency of 1, or 100 percent, would be possible only if there were no heat to the environment (
).
### Solving Thermal Efficiency Problems
### Practice Problems
### Check Your Understanding
### Section Summary
1. Heat engines use the heat transfer of energy to do work.
2. Cyclical processes are processes that return to their original state at the end of every cycle.
3. The thermal efficiency of a heat engine is the ratio of work output divided by the amount of energy input.
4. The amount of work a heat engine can do is determined by the net heat transfer of energy during a cycle; more waste heat leads to less work output.
5. Heat pumps draw energy by heat from cold outside air and use it to heat an interior room.
6. A refrigerator is a type of heat pump; it takes energy from the warm air from the inside compartment and transfers it to warmer exterior air.
### Key Equations
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# Waves and Their Properties
## Introduction
Recall from the chapter on Motion in Two Dimensions that oscillations—the back-and-forth movement between two points—involve force and energy. Some oscillations create waves, such as the sound waves created by plucking a guitar string. Other examples of waves include earthquakes and visible light. Even subatomic particles, such as electrons, can behave like waves. You can make water waves in a swimming pool by slapping the water with your hand. Some of these waves, such as water waves, are visible; others, such as sound waves, are not. But every wave is a disturbance that moves from its source and carries energy. In this chapter, we will learn about the different types of waves, their properties, and how they interact with one another. |
# Waves and Their Properties
## Types of Waves
### Section Key Terms
### Mechanical Waves
What do we mean when we say something is a wave? A wave is a disturbance that travels or propagates from the place where it was created. Waves transfer energy from one place to another, but they do not necessarily transfer any mass. Light, sound, and waves in the ocean are common examples of waves. Sound and water waves are mechanical waves; meaning, they require a medium to travel through. The medium may be a solid, a liquid, or a gas, and the speed of the wave depends on the material properties of the medium through which it is traveling. However, light is not a mechanical wave; it can travel through a vacuum such as the empty parts of outer space.
A familiar wave that you can easily imagine is the water wave. For water waves, the disturbance is in the surface of the water, an example of which is the disturbance created by a rock thrown into a pond or by a swimmer splashing the water surface repeatedly. For sound waves, the disturbance is caused by a change in air pressure, an example of which is when the oscillating cone inside a speaker creates a disturbance. For earthquakes, there are several types of disturbances, which include the disturbance of Earth’s surface itself and the pressure disturbances under the surface. Even radio waves are most easily understood using an analogy with water waves. Because water waves are common and visible, visualizing water waves may help you in studying other types of waves, especially those that are not visible.
Water waves have characteristics common to all waves, such as amplitude, period, frequency, and energy, which we will discuss in the next section.
### Pulse Waves and Periodic Waves
If you drop a pebble into the water, only a few waves may be generated before the disturbance dies down, whereas in a wave pool, the waves are continuous. A pulse wave is a sudden disturbance in which only one wave or a few waves are generated, such as in the example of the pebble. Thunder and explosions also create pulse waves. A periodic wave repeats the same oscillation for several cycles, such as in the case of the wave pool, and is associated with simple harmonic motion. Each particle in the medium experiences simple harmonic motion in periodic waves by moving back and forth periodically through the same positions.
Consider the simplified water wave in . This wave is an up-and-down disturbance of the water surface, characterized by a sine wave pattern. The uppermost position is called the crest and the lowest is the trough. It causes a seagull to move up and down in simple harmonic motion as the wave crests and troughs pass under the bird.
### Longitudinal Waves and Transverse Waves
Mechanical waves are categorized by their type of motion and fall into any of two categories: transverse or longitudinal. Note that both transverse and longitudinal waves can be periodic. A transverse wave propagates so that the disturbance is perpendicular to the direction of propagation. An example of a transverse wave is shown in , where a woman moves a toy spring up and down, generating waves that propagate away from herself in the horizontal direction while disturbing the toy spring in the vertical direction.
In contrast, in a longitudinal wave, the disturbance is parallel to the direction of propagation. shows an example of a longitudinal wave, where the woman now creates a disturbance in the horizontal direction—which is the same direction as the wave propagation—by stretching and then compressing the toy spring.
Waves may be transverse, longitudinal, or a combination of the two. The waves on the strings of musical instruments are transverse (as shown in ), and so are electromagnetic waves, such as visible light. Sound waves in air and water are longitudinal. Their disturbances are periodic variations in pressure that are transmitted in fluids.
Sound in solids can be both longitudinal and transverse. Essentially, water waves are also a combination of transverse and longitudinal components, although the simplified water wave illustrated in does not show the longitudinal motion of the bird.
Earthquake waves under Earth’s surface have both longitudinal and transverse components as well. The longitudinal waves in an earthquake are called pressure or P-waves, and the transverse waves are called shear or S-waves. These components have important individual characteristics; for example, they propagate at different speeds. Earthquakes also have surface waves that are similar to surface waves on water.
### Check Your Understanding
### Section Summary
1. A wave is a disturbance that moves from the point of creation and carries energy but not mass.
2. Mechanical waves must travel through a medium.
3. Sound waves, water waves, and earthquake waves are all examples of mechanical waves.
4. Light is not a mechanical wave since it can travel through a vacuum.
5. A periodic wave is a wave that repeats for several cycles, whereas a pulse wave has only one crest or a few crests and is associated with a sudden disturbance.
6. Periodic waves are associated with simple harmonic motion.
7. A transverse wave has a disturbance perpendicular to its direction of propagation, whereas a longitudinal wave has a disturbance parallel to its direction of propagation.
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# Waves and Their Properties
## Wave Properties: Speed, Amplitude, Frequency, and Period
### Section Key Terms
### Wave Variables
In the chapter on motion in two dimensions, we defined the following variables to describe harmonic motion:
1. Amplitude—maximum displacement from the equilibrium position of an object oscillating around such equilibrium position
2. Frequency—number of events per unit of time
3. Period—time it takes to complete one oscillation
For waves, these variables have the same basic meaning. However, it is helpful to word the definitions in a more specific way that applies directly to waves:
1. Amplitude—distance between the resting position and the maximum displacement of the wave
2. Frequency—number of waves passing by a specific point per second
3. Period—time it takes for one wave cycle to complete
In addition to amplitude, frequency, and period, their wavelength and wave velocity also characterize waves. The wavelength
is the distance between adjacent identical parts of a wave, parallel to the direction of propagation. The wave velocity
is the speed at which the disturbance moves.
Consider the periodic water wave in . Its wavelength is the distance from crest to crest or from trough to trough. The wavelength can also be thought of as the distance a wave has traveled after one complete cycle—or one period. The time for one complete up-and-down motion is the simple water wave’s period T. In the figure, the wave itself moves to the right with a wave velocity vw. Its amplitude X is the distance between the resting position and the maximum displacement—either the crest or the trough—of the wave. It is important to note that this movement of the wave is actually the disturbance moving to the right, not the water itself; otherwise, the bird would move to the right. Instead, the seagull bobs up and down in place as waves pass underneath, traveling a total distance of 2X in one cycle. However, as mentioned in the text feature on surfing, actual ocean waves are more complex than this simplified example.
### The Relationship between Wave Frequency, Period, Wavelength, and Velocity
Since wave frequency is the number of waves per second, and the period is essentially the number of seconds per wave, the relationship between frequency and period is
or
just as in the case of harmonic motion of an object. We can see from this relationship that a higher frequency means a shorter period. Recall that the unit for frequency is hertz (Hz), and that 1 Hz is one cycle—or one wave—per second.
The speed of propagation vw is the distance the wave travels in a given time, which is one wavelength in a time of one period. In equation form, it is written as
or
From this relationship, we see that in a medium where vw is constant, the higher the frequency, the smaller the wavelength. See .
These fundamental relationships hold true for all types of waves. As an example, for water waves, vw is the speed of a surface wave; for sound, vw is the speed of sound; and for visible light, vw is the speed of light. The amplitude X is completely independent of the speed of propagation vw and depends only on the amount of energy in the wave.
### Solving Wave Problems
### Practice Problems
### Check Your Understanding
### Section Summary
1. A wave is a disturbance that moves from the point of creation at a wave velocity vw.
2. A wave has a wavelength
, which is the distance between adjacent identical parts of the wave.
3. The wave velocity and the wavelength are related to the wave’s frequency and period by
or
4. The time for one complete wave cycle is the period T.
5. The number of waves per unit time is the frequency ƒ.
6. The wave frequency and the period are inversely related to one another.
### Key Equations
### Concept Items
### Critical Thinking
### Problems
### Test Prep Multiple Choice
### Test Prep Short Answer
### Test Prep Extended Response
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# Waves and Their Properties
## Wave Interaction: Superposition and Interference
### Section Key Terms
### Superposition of Waves
Most waves do not look very simple. They look more like the waves in , rather than the simple water wave considered in the previous sections, which has a perfect sinusoidal shape.
Most waves appear complex because they result from two or more simple waves that combine as they come together at the same place at the same time—a phenomenon called superposition.
Waves superimpose by adding their disturbances; each disturbance corresponds to a force, and all the forces add. If the disturbances are along the same line, then the resulting wave is a simple addition of the disturbances of the individual waves, that is, their amplitudes add.
### Wave Interference
The two special cases of superposition that produce the simplest results are pure constructive interference and pure destructive interference.
Pure constructive interference occurs when two identical waves arrive at the same point exactly in phase. When waves are exactly in phase, the crests of the two waves are precisely aligned, as are the troughs. Refer to . Because the disturbances add, the pure constructive interference of two waves with the same amplitude produces a wave that has twice the amplitude of the two individual waves, but has the same wavelength.
shows two identical waves that arrive exactly out of phase—that is, precisely aligned crest to trough—producing pure destructive interference. Because the disturbances are in opposite directions for this superposition, the resulting amplitude is zero for pure destructive interference; that is, the waves completely cancel out each other.
While pure constructive interference and pure destructive interference can occur, they are not very common because they require precisely aligned identical waves. The superposition of most waves that we see in nature produces a combination of constructive and destructive interferences.
Waves that are not results of pure constructive or destructive interference can vary from place to place and time to time. The sound from a stereo, for example, can be loud in one spot and soft in another. The varying loudness means that the sound waves add partially constructively and partially destructively at different locations. A stereo has at least two speakers that create sound waves, and waves can reflect from walls. All these waves superimpose.
An example of sounds that vary over time from constructive to destructive is found in the combined whine of jet engines heard by a stationary passenger. The volume of the combined sound can fluctuate up and down as the sound from the two engines varies in time from constructive to destructive.
The two previous examples considered waves that are similar—both stereo speakers generate sound waves with the same amplitude and wavelength, as do the jet engines. But what happens when two waves that are not similar, that is, having different amplitudes and wavelengths, are superimposed? An example of the superposition of two dissimilar waves is shown in . Here again, the disturbances add and subtract, but they produce an even more complicated-looking wave. The resultant wave from the combined disturbances of two dissimilar waves looks much different than the idealized sinusoidal shape of a periodic wave.
### Standing Waves
Sometimes waves do not seem to move and they appear to just stand in place, vibrating. Such waves are called standing waves and are formed by the superposition of two or more waves moving in opposite directions. The waves move through each other with their disturbances adding as they go by. If the two waves have the same amplitude and wavelength, then they alternate between constructive and destructive interference. Standing waves created by the superposition of two identical waves moving in opposite directions are illustrated in .
As an example, standing waves can be seen on the surface of a glass of milk in a refrigerator. The vibrations from the refrigerator motor create waves on the milk that oscillate up and down but do not seem to move across the surface. The two waves that produce standing waves may be due to the reflections from the side of the glass.
Earthquakes can create standing waves and cause constructive and destructive interferences. As the earthquake waves travel along the surface of Earth and reflect off denser rocks, constructive interference occurs at certain points. As a result, areas closer to the epicenter are not damaged while areas farther from the epicenter are damaged.
Standing waves are also found on the strings of musical instruments and are due to reflections of waves from the ends of the string. and show three standing waves that can be created on a string that is fixed at both ends. When the wave reaches the fixed end, it has nowhere else to go but back where it came from, causing the reflection. The nodes are the points where the string does not move; more generally, the nodes are the points where the wave disturbance is zero in a standing wave. The fixed ends of strings must be nodes, too, because the string cannot move there.
The antinode is the location of maximum amplitude in standing waves. The standing waves on a string have a frequency that is related to the propagation speed
of the disturbance on the string. The wavelength
is determined by the distance between the points where the string is fixed in place.
### Reflection and Refraction of Waves
As we saw in the case of standing waves on the strings of a musical instrument, reflection is the change in direction of a wave when it bounces off a barrier, such as a fixed end. When the wave hits the fixed end, it changes direction, returning to its source. As it is reflected, the wave experiences an inversion, which means that it flips vertically. If a wave hits the fixed end with a crest, it will return as a trough, and vice versa (Henderson 2015). Refer to .
Rather than encountering a fixed end or barrier, waves sometimes pass from one medium into another, for instance, from air into water. Different types of media have different properties, such as density or depth, that affect how a wave travels through them. At the boundary between media, waves experience refraction—they change their path of propagation. As the wave bends, it also changes its speed and wavelength upon entering the new medium. Refer to .
For example, water waves traveling from the deep end to the shallow end of a swimming pool experience refraction. They bend in a path closer to perpendicular to the surface of the water, propagate slower, and decrease in wavelength as they enter shallower water.
### Check Your Understanding
### Section Summary
1. Superposition is the combination of two waves at the same location.
2. Constructive interference occurs when two identical waves are superimposed exactly in phase.
3. Destructive interference occurs when two identical waves are superimposed exactly out of phase.
4. A standing wave is a wave produced by the superposition of two waves. It varies in amplitude but does not propagate.
5. The nodes are the points where there is no motion in standing waves.
6. An antinode is the location of maximum amplitude of a standing wave.
7. Reflection causes a wave to change direction.
8. Inversion occurs when a wave reflects from a fixed end.
9. Refraction causes a wave’s path to bend and occurs when a wave passes from one medium into another medium with a different density.
### Concept Items
### Critical Thinking
### Test Prep Multiple Choice
### Test Prep Short Answer
### Performance Task
### Test Prep Extended Response
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