Preface

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About *University Physics*

*University Physics* is designed for the two- or three-semester calculus-based physics course. The text has been developed to meet the scope and sequence of most university physics courses and provides a foundation for a career in mathematics, science, or engineering. The book provides an important opportunity for students to learn the core concepts of physics and understand how those concepts apply to their lives and to the world around them.

Additional resources

About the authors

Mechanics

Units and Measurement

Introduction

The Scope and Scale of Physics

Abstract:
By the end of this section, you will be able to:

* Describe the scope of physics.
* Calculate the order of magnitude of a quantity.
* Compare measurable length, mass, and timescales quantitatively.
* Describe the relationships among models, theories, and laws.

The Scope of Physics

Take another look at the chapter-opening image. The Whirlpool Galaxy contains billions of individual stars as well as huge clouds of gas and dust. Its companion galaxy is also visible to the right. This pair of galaxies lies a staggering billion trillion miles $\left(1.4{\rule{0.2em}{0ex}}×{\rule{0.2em}{0ex}}{10}^{21}\text{mi}\right)$ from our own galaxy (which is called the *Milky Way*). The stars and planets that make up the Whirlpool Galaxy might seem to be the furthest thing from most people’s everyday lives, but the Whirlpool is a great starting point to think about the forces that hold the universe together. The forces that cause the Whirlpool Galaxy to act as it does are thought to be the same forces we contend with here on Earth, whether we are planning to send a rocket into space or simply planning to raise the walls for a new home. The gravity that causes the stars of the Whirlpool Galaxy to rotate and revolve is thought to be the same as what causes water to flow over hydroelectric dams here on Earth. When you look up at the stars, realize the forces out there are the same as the ones here on Earth. Through a study of physics, you may gain a greater understanding of the interconnectedness of everything we can see and know in this universe.

The Scale of Physics

From the discussion so far, it should be clear that to accomplish your goals in any of the various fields within the natural sciences and engineering, a thorough grounding in the laws of physics is necessary. The reason for this is simply that the laws of physics govern everything in the observable universe at all measurable scales of length, mass, and time. Now, that is easy enough to say, but to come to grips with what it really means, we need to get a little bit quantitative. So, before surveying the various scales that physics allows us to explore, let’s first look at the concept of “order of magnitude,” which we use to come to terms with the vast ranges of length, mass, and time that we consider in this text ([link]).

Building Models

How did we come to know the laws governing natural phenomena? What we refer to as the laws of nature are concise descriptions of the universe around us. They are human statements of the underlying laws or rules that all natural processes follow. Such laws are intrinsic to the universe; 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 ([link]). The cornerstone of discovering natural laws is observation; scientists must describe the universe as it is, not as we imagine it to be.

Units and Standards

Abstract:
By the end of this section, you will be able to:

* Describe how SI base units are defined.
* Describe how derived units are created from base units.
* Express quantities given in SI units using metric prefixes.

SI Units: Base 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 unit**s**. 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.

Units of Time, Length, and Mass: The Second, Meter, and Kilogram

The initial chapters in this textbook are concerned with mechanics, fluids, and waves. In these subjects all pertinent physical quantities can be expressed in terms of the base units of length, mass, and time. Therefore, we now turn to a discussion of these three base units, leaving discussion of the others until they are needed later.

Metric Prefixes

SI units are part of the metric system, which is convenient for scientific and engineering calculations because the units are categorized by factors of 10. [link] lists the metric prefixes and symbols used to denote various factors of 10 in SI units. For example, a centimeter is one-hundredth of a meter (in symbols, 1 cm = 10–2 m) and a kilometer is a thousand meters (1 km = 103 m). Similarly, a megagram is a million grams (1 Mg = 106 g), a nanosecond is a billionth of a second (1 ns = 10–9 s), and a terameter is a trillion meters (1 Tm = 1012 m).

Unit Conversion

Abstract:
By the end of this section, you will be able to:

* Use conversion factors to express the value of a given quantity in different units.

Dimensional Analysis

Abstract:
By the end of this section, you will be able to:

* Find the dimensions of a mathematical expression involving physical quantities.
* Determine whether an equation involving physical quantities is dimensionally consistent.

Estimates and Fermi Calculations

Abstract:
By the end of this section, you will be able to:

* Estimate the values of physical quantities.

Significant Figures

Abstract:
By the end of this section, you will be able to:

* Determine the correct number of significant figures for the result of a computation.
* Describe the relationship between the concepts of accuracy, precision, uncertainty, and discrepancy.
* Calculate the percent uncertainty of a measurement, given its value and its uncertainty.
* Determine the uncertainty of the result of a computation involving quantities with given uncertainties.

Accuracy and Precision of a Measurement

Science is based on observation and experiment—that is, on measurements. **Accuracy** is how close a measurement is to the accepted reference value for that measurement. For example, let’s say we want to measure the length of standard printer paper. The packaging in which we purchased the paper states that it is 11.0 in. long. We then measure the length of the paper three times and obtain the following measurements: 11.1 in., 11.2 in., and 10.9 in. These measurements are quite accurate because they are very close to the reference value of 11.0 in. In contrast, if we had obtained a measurement of 12 in., our measurement would not be very accurate. Notice that the concept of accuracy requires that an accepted reference value be given.

Accuracy, Precision, Uncertainty, and Discrepancy

The precision of a measuring system is related to the uncertainty in the measurements whereas the accuracy is related to the discrepancy from the accepted reference value. Uncertainty is a quantitative measure of how much your measured values deviate from one another. There are many different methods of calculating uncertainty, each of which is appropriate to different situations. Some examples include taking the range (that is, the largest minus the smallest) or finding the standard deviation of the measurements. Discrepancy (or “measurement error”) is the difference between the measured value and a given standard or expected value. If the measurements are not very precise, then the uncertainty of the values is high. If the measurements are not very accurate, then the discrepancy of the values is high.

Precision of Measuring Tools and Significant Figures

An important factor in the precision of measurements involves the precision of the measuring tool. In general, a precise measuring tool is one that can measure values in very small increments. For example, a standard ruler can measure length to the nearest millimeter whereas a caliper can measure length to the nearest 0.01 mm. The caliper is a more precise measuring tool because it can measure extremely small differences in length. The more precise the measuring tool, the more precise the measurements.

Solving Problems in Physics

Abstract:
By the end of this section, you will be able to:

* Describe the process for developing a problem-solving strategy.
* Explain how to find the numerical solution to a problem.
* Summarize the process for assessing the significance of the numerical solution to a problem.

Strategy

Strategy is the beginning stage of solving a problem. The idea is to figure out exactly what the problem is and then develop a strategy for solving it. Some general advice for this stage is as follows:

Solution

The solution stage is when you do the math. *Substitute the knowns (along with their units) into the appropriate equation and obtain numerical solutions complete with units*. That is, do the algebra, calculus, geometry, or arithmetic necessary to find the unknown from the knowns, being sure to carry the units through the calculations. This step is clearly important because it produces the numerical answer, along with its units. Notice, however, that this stage is only one-third of the overall problem-solving process.

Significance

After having done the math in the solution stage of problem solving, it is tempting to think you are done. But, always remember that physics is not math. Rather, in doing physics, we use mathematics as a tool to help us understand nature. So, after you obtain a numerical answer, you should always assess its significance:

Vectors

Introduction

Scalars and Vectors

Abstract:
By the end of this section, you will be able to:

* Describe the difference between vector and scalar quantities.
* Identify the magnitude and direction of a vector.
* Explain the effect of multiplying a vector quantity by a scalar.
* Describe how one-dimensional vector quantities are added or subtracted.
* Explain the geometric construction for the addition or subtraction of vectors in a plane.
* Distinguish between a vector equation and a scalar equation.

Algebra of Vectors in One Dimension

Vectors can be multiplied by scalars, added to other vectors, or subtracted from other vectors. We can illustrate these vector concepts using an example of the fishing trip seen in [link].

Algebra of Vectors in Two Dimensions

When vectors lie in a plane—that is, when they are in two dimensions—they can be multiplied by scalars, added to other vectors, or subtracted from other vectors in accordance with the general laws expressed by [link], [link], [link], and [link]. However, the addition rule for two vectors in a plane becomes more complicated than the rule for vector addition in one dimension. We have to use the laws of geometry to construct resultant vectors, followed by trigonometry to find vector magnitudes and directions. This geometric approach is commonly used in navigation ([link]). In this section, we need to have at hand two rulers, a triangle, a protractor, a pencil, and an eraser for drawing vectors to scale by geometric constructions.

Coordinate Systems and Components of a Vector

Abstract:
By the end of this section, you will be able to:

* Describe vectors in two and three dimensions in terms of their components, using unit vectors along the axes.
* Distinguish between the vector components of a vector and the scalar components of a vector.
* Explain how the magnitude of a vector is defined in terms of the components of a vector.
* Identify the direction angle of a vector in a plane.
* Explain the connection between polar coordinates and Cartesian coordinates in a plane.

Polar Coordinates

To describe locations of points or vectors in a plane, we need two orthogonal directions. In the Cartesian coordinate system these directions are given by unit vectors $\stackrel{^}{i}$ and $\stackrel{^}{j}$ along the *x*-axis and the *y*-axis, respectively. The Cartesian coordinate system is very convenient to use in describing displacements and velocities of objects and the forces acting on them. However, it becomes cumbersome when we need to describe the rotation of objects. When describing rotation, we usually work in the polar coordinate system.

Vectors in Three Dimensions

To specify the location of a point in space, we need three coordinates (*x*, *y*, *z*), where coordinates *x* and *y* specify locations in a plane, and coordinate *z* gives a vertical position above or below the plane. Three-dimensional space has three orthogonal directions, so we need not two but *three* unit vectors to define a three-dimensional coordinate system. In the Cartesian coordinate system, the first two unit vectors are the unit vector of the *x*-axis $\stackrel{^}{i}$ and the unit vector of the *y*-axis $\stackrel{^}{j}$. The third unit vector $\stackrel{^}{k}$ is the direction of the *z*-axis ([link]). The order in which the axes are labeled, which is the order in which the three unit vectors appear, is important because it defines the orientation of the coordinate system. The order *x*-*y*-*z*, which is equivalent to the order $\stackrel{^}{i}$ - $\stackrel{^}{j}$ - $\stackrel{^}{k}$, defines the standard right-handed coordinate system (positive orientation). Notice that while previously the *y*-axis was used for the upward direction, in three dimensions we usually choose *z* to represent the vertical. A right-handed coordinate system with *z* up and with one of the remaining axes pointing to the right and the other out of the page must have the *y*-axis pointing right and *x* out of the page, as illustrated here.

Algebra of Vectors

Abstract:
By the end of this section, you will be able to:

* Apply analytical methods of vector algebra to find resultant vectors and to solve vector equations for unknown vectors.
* Interpret physical situations in terms of vector expressions.

Products of Vectors

Abstract:
By the end of this section, you will be able to:

* Explain the difference between the scalar product and the vector product of two vectors.
* Determine the scalar product of two vectors.
* Determine the vector product of two vectors.
* Describe how the products of vectors are used in physics.

The Scalar Product of Two Vectors (the Dot Product)

Scalar multiplication of two vectors yields a scalar product.

The Vector Product of Two Vectors (the Cross Product)

Vector multiplication of two vectors yields a vector product.

Motion Along a Straight Line

Introduction

Position, Displacement, and Average Velocity

Abstract:
By the end of this section, you will be able to:

* Define position, displacement, and distance traveled.
* Calculate the total displacement given the position as a function of time.
* Determine the total distance traveled.
* Calculate the average velocity given the displacement and elapsed time.

Position

To describe the motion of an object, you must first be able to describe its position (*x*): *where it is at any particular time*. More precisely, we need to specify its position relative to a convenient frame of reference. A frame of reference is an arbitrary set of axes from which the position and motion of an object are described. Earth is often used as a frame of reference, and we often describe the position of an object as it relates to stationary objects on Earth. For example, a rocket launch could be described in terms of the position of the rocket with respect to Earth as a whole, whereas a cyclist’s position could be described in terms of where she is in relation to the buildings she passes [link]. 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. To describe the position of an object undergoing one-dimensional motion, we often use the variable *x*. Later in the chapter, during the discussion of free fall, we use the variable *y*.

Displacement

If an object moves relative to a frame of reference—for example, if a professor moves to the right relative to a whiteboard [link]—then the object’s position changes. This change in position is called displacement. The word *displacement* implies that an object has moved, or has been displaced. Although position is the numerical value of *x* along a straight line where an object might be located, displacement gives the *change* in position along this line. Since displacement indicates direction, it is a vector and can be either positive or negative, depending on the choice of positive direction. Also, an analysis of motion can have many displacements embedded in it. If right is positive and an object moves 2 m to the right, then 4 m to the left, the individual displacements are 2 m and $-4$ m, respectively.

Average Velocity

To calculate the other physical quantities in kinematics we must introduce the time variable. The time variable allows us not only to state where the object is (its position) during its motion, but also how fast it is moving. How fast an object is moving is given by the rate at which the position changes with time.

Instantaneous Velocity and Speed

Abstract:
By the end of this section, you will be able to:

* Explain the difference between average velocity and instantaneous velocity.
* Describe the difference between velocity and speed.
* Calculate the instantaneous velocity given the mathematical equation for the position.
* Calculate the speed given the instantaneous velocity.

Instantaneous Velocity

The quantity that tells us how fast an object is moving anywhere along its path is the instantaneous velocity, usually called simply *velocity*. It is the average velocity between two points on the path in the limit that the time (and therefore the displacement) between the two events approaches zero. To illustrate this idea mathematically, we need to express position *x* as a continuous function of *t* denoted by *x*(*t*). The expression for the average velocity between two points using this notation is $\stackrel{\text{–}}{v}=\frac{x\left({t}_{2}\right)-x\left({t}_{1}\right)}{{t}_{2}-{t}_{1}}$. To find the instantaneous velocity at any position, we let ${t}_{1}=t$ and ${t}_{2}=t+\text{Δ}t$. After inserting these expressions into the equation for the average velocity and taking the limit as $\text{Δ}t\to 0$, we find the expression for the instantaneous velocity:

Speed

In everyday language, most people use the terms *speed* and *velocity* interchangeably. In physics, however, they do not have the same meaning and are distinct concepts. One major difference is that speed has no direction; that is, speed is a scalar.

Calculating Instantaneous Velocity

When calculating instantaneous velocity, we need to specify the explicit form of the position function $x\left(t\right)$. If each term in the $x\left(t\right)$ equation has the form of $A{t}^{n}$ where $A$ is a constant and $n$ is an integer, this can be differentiated using the power rule to be:

Average and Instantaneous Acceleration

Abstract:
By the end of this section, you will be able to:

* Calculate the average acceleration between two points in time.
* Calculate the instantaneous acceleration given the functional form of velocity.
* Explain the vector nature of instantaneous acceleration and velocity.
* Explain the difference between average acceleration and instantaneous acceleration.
* Find instantaneous acceleration at a specified time on a graph of velocity versus time.

Average Acceleration

The formal definition of acceleration is consistent with these notions just described, but is more inclusive.

Instantaneous Acceleration

Instantaneous acceleration *a*, or *acceleration at a specific instant in time*, is obtained using the same process discussed for instantaneous velocity. That is, we calculate the average acceleration between two points in time separated by $\text{Δ}t$ and let $\text{Δ}t$ approach zero. The result is the derivative of the velocity function *v*(*t*), which is instantaneous acceleration and is expressed mathematically as

Getting a Feel for Acceleration

You are probably used to experiencing acceleration when you step into an elevator, or step on the gas pedal in your car. However, acceleration is happening to many other objects in our universe with which we don’t have direct contact. [link] presents the acceleration of various objects. We can see the magnitudes of the accelerations extend over many orders of magnitude.

Motion with Constant Acceleration

Abstract:
By the end of this section, you will be able to:

* Identify which equations of motion are to be used to solve for unknowns.
* Use appropriate equations of motion to solve a two-body pursuit problem.

Notation

First, let us make some simplifications in notation. Taking the initial time to be zero, as if time is measured with a stopwatch, is a great simplification. Since elapsed time is $\text{Δ}t={t}_{\text{f}}-{t}_{0}$, taking ${t}_{0}=0$ means that$\text{Δ}t={t}_{\text{f}}$, the final time on the stopwatch. When initial time is taken to be zero, we use the subscript 0 to denote initial values of position and velocity. That is, ${x}_{0}$ *is the initial position* and ${v}_{0}$ *is the initial velocity*. We put no subscripts on the final values. That is, *t is the final time*, *x is the final position*, and *v is the final velocity*. This gives a simpler expression for elapsed time, $\text{Δ}t=t$. It also simplifies the expression for *x* displacement, which is now $\text{Δ}x=x-{x}_{0}$. Also, it simplifies the expression for change in velocity, which is now $\text{Δ}v=v-{v}_{0}$. To summarize, using the simplified notation, with the initial time taken to be zero,

Displacement and Position from Velocity

To get our first two equations, we start with the definition of average velocity:

Solving for Final Velocity from Acceleration and Time

We can derive another useful equation by manipulating the definition of acceleration:

Solving for Final Position with Constant Acceleration

We can combine the previous equations to find a third equation that allows us to calculate the final position of an object experiencing constant acceleration. We start with

Solving for Final Velocity from Distance and Acceleration

A fourth useful equation can be obtained from another algebraic manipulation of previous equations. If we solve $v={v}_{0}+at$ for *t*, we get

Putting Equations Together

In the following examples, we continue to explore one-dimensional motion, but in situations requiring slightly more algebraic manipulation. The examples also give insight into problem-solving techniques. The note that follows is provided for easy reference to the equations needed. Be aware that these equations are not independent. In many situations we have two unknowns and need two equations from the set to solve for the unknowns. We need as many equations as there are unknowns to solve a given situation.

Two-Body Pursuit Problems

Up until this point we have looked at examples of motion involving a single body. Even for the problem with two cars and the stopping distances on wet and dry roads, we divided this problem into two separate problems to find the answers. In a two-body pursuit problem, the motions of the objects are coupled—meaning, the unknown we seek depends on the motion of both objects. To solve these problems we write the equations of motion for each object and then solve them simultaneously to find the unknown. This is illustrated in [link].

Free Fall

Abstract:
By the end of this section, you will be able to:

* Use the kinematic equations with the variables y and g to analyze free-fall motion.
* Describe how the values of the position, velocity, and acceleration change during a free fall.
* Solve for the position, velocity, and acceleration as functions of time when an object is in a free fall.

Gravity

The most remarkable and unexpected fact about falling objects is that if air resistance and friction are negligible, then in a given location all objects fall toward the center of Earth with the *same constant acceleration*, *independent of their mass*. This experimentally determined fact is unexpected because we are so accustomed to the effects of air resistance and friction that we expect light objects to fall slower than heavy ones. Until Galileo Galilei (1564–1642) proved otherwise, people believed that a heavier object has a greater acceleration in a free fall. We now know this is not the case. In the absence of air resistance, heavy objects arrive at the ground at the same time as lighter objects when dropped from the same height [link].

One-Dimensional Motion Involving Gravity

The best way to see the basic features of motion involving gravity is to start with the simplest situations and then progress toward more complex ones. So, we start by considering straight up-and-down motion with no air resistance or friction. These assumptions mean the velocity (if there is any) is vertical. If an object is dropped, we know the initial velocity is zero when in free fall. When the object has left contact with whatever held or threw it, the object is in free fall. When the object is thrown, it has the same initial speed in free fall as it did before it was released. When the object comes in contact with the ground or any other object, it is no longer in free fall and its acceleration of *g* is no longer valid. Under these circumstances, the motion is one-dimensional and has constant acceleration of magnitude *g*. We represent vertical displacement with the symbol *y*.

Finding Velocity and Displacement from Acceleration

Abstract:
By the end of this section, you will be able to:

* Derive the kinematic equations for constant acceleration using integral calculus.
* Use the integral formulation of the kinematic equations in analyzing motion.
* Find the functional form of velocity versus time given the acceleration function.
* Find the functional form of position versus time given the velocity function.

Kinematic Equations from Integral Calculus

Let’s begin with a particle with an acceleration *a*(t) which is a known function of time. Since the time derivative of the velocity function is acceleration,

Motion in Two and Three Dimensions

Introduction

Displacement and Velocity Vectors

Abstract:
By the end of this section, you will be able to:

* Calculate position vectors in a multidimensional displacement problem.
* Solve for the displacement in two or three dimensions.
* Calculate the velocity vector given the position vector as a function of time.
* Calculate the average velocity in multiple dimensions.

Displacement Vector

To describe motion in two and three dimensions, we must first establish a coordinate system and a convention for the axes. We generally use the coordinates *x*, *y*, and *z* to locate a particle at point *P*(*x*, *y*, *z*) in three dimensions. If the particle is moving, the variables *x*, *y*, and *z* are functions of time (*t*):

Velocity Vector

In the previous chapter we found the instantaneous velocity by calculating the derivative of the position function with respect to time. We can do the same operation in two and three dimensions, but we use vectors. The instantaneous velocity vector is now

The Independence of Perpendicular Motions

When we look at the three-dimensional equations for position and velocity written in unit vector notation, [link] and [link], we see the components of these equations are separate and unique functions of time that do not depend on one another. Motion along the *x* direction has no part of its motion along the *y* and *z* directions, and similarly for the other two coordinate axes. Thus, the motion of an object in two or three dimensions can be divided into separate, independent motions along the perpendicular axes of the coordinate system in which the motion takes place.

Acceleration Vector

Abstract:
By the end of this section, you will be able to:

* Calculate the acceleration vector given the velocity function in unit vector notation.
* Describe the motion of a particle with a constant acceleration in three dimensions.
* Use the one-dimensional motion equations along perpendicular axes to solve a problem in two or three dimensions with a constant acceleration.
* Express the acceleration in unit vector notation.

Instantaneous Acceleration

In addition to obtaining the displacement and velocity vectors of an object in motion, we often want to know its acceleration vector at any point in time along its trajectory. This acceleration vector is the instantaneous acceleration and it can be obtained from the derivative with respect to time of the velocity function, as we have seen in a previous chapter. The only difference in two or three dimensions is that these are now vector quantities. Taking the derivative with respect to time $\stackrel{\to }{v}\left(t\right),$ we find

Constant Acceleration

Multidimensional motion with constant acceleration can be treated the same way as shown in the previous chapter for one-dimensional motion. Earlier we showed that three-dimensional motion is equivalent to three one-dimensional motions, each along an axis perpendicular to the others. To develop the relevant equations in each direction, let’s consider the two-dimensional problem of a particle moving in the *xy* plane with constant acceleration, ignoring the *z*-component for the moment. The acceleration vector is

Projectile Motion

Abstract:
By the end of this section, you will be able to:

* Use one-dimensional motion in perpendicular directions to analyze projectile motion.
* Calculate the range, time of flight, and maximum height of a projectile that is launched and impacts a flat, horizontal surface.
* Find the time of flight and impact velocity of a projectile that lands at a different height from that of launch.
* Calculate the trajectory of a projectile.

Time of Flight, Trajectory, and Range

Of interest are the time of flight, trajectory, and range for a projectile launched on a flat horizontal surface and impacting on the same surface. In this case, kinematic equations give useful expressions for these quantities, which are derived in the following sections.

Uniform Circular Motion

Abstract:
By the end of this section, you will be able to:

* Solve for the centripetal acceleration of an object moving on a circular path.
* Use the equations of circular motion to find the position, velocity, and acceleration of a particle executing circular motion.
* Explain the differences between centripetal acceleration and tangential acceleration resulting from nonuniform circular motion.
* Evaluate centripetal and tangential acceleration in nonuniform circular motion, and find the total acceleration vector.

Centripetal Acceleration

In one-dimensional kinematics, objects with a constant speed have zero acceleration. However, in two- and three-dimensional kinematics, even if the speed is a constant, a particle can have acceleration if it moves along a curved trajectory such as a circle. In this case the velocity vector is changing, or $d\stackrel{\to }{v}\text{/}dt\ne 0.$ This is shown in [link]. As the particle moves counterclockwise in time $\text{Δ}t$ on the circular path, its position vector moves from $\stackrel{\to }{r}\left(t\right)$ to $\stackrel{\to }{r}\left(t+\text{Δ}t\right).$ The velocity vector has constant magnitude and is tangent to the path as it changes from $\stackrel{\to }{v}\left(t\right)$ to $\stackrel{\to }{v}\left(t+\text{Δ}t\right),$ changing its direction only. Since the velocity vector $\stackrel{\to }{v}\left(t\right)$ is perpendicular to the position vector $\stackrel{\to }{r}\left(t\right),$ the triangles formed by the position vectors and $\text{Δ}\stackrel{\to }{r},$ and the velocity vectors and $\text{Δ}\stackrel{\to }{v}$ are similar. Furthermore, since $|\stackrel{\to }{r}\left(t\right)|=|\stackrel{\to }{r}\left(t+\text{Δ}t\right)|$ and $|\stackrel{\to }{v}\left(t\right)|=|\stackrel{\to }{v}\left(t+\text{Δ}t\right)|,$ the two triangles are isosceles. From these facts we can make the assertion

Equations of Motion for Uniform Circular Motion

A particle executing circular motion can be described by its position vector $\stackrel{\to }{r}\left(t\right).$ [link] shows a particle executing circular motion in a counterclockwise direction. As the particle moves on the circle, its position vector sweeps out the angle $\theta$ with the *x-*axis. Vector $\stackrel{\to }{r}\left(t\right)$ making an angle $\theta$ with the *x-*axis is shown with its components along the *x*- and *y*-axes. The magnitude of the position vector is $A=|\stackrel{\to }{r}\left(t\right)|$ and is also the radius of the circle, so that in terms of its components,

Nonuniform Circular Motion

Circular motion does not have to be at a constant speed. A particle can travel in a circle and speed up or slow down, showing an acceleration in the direction of the motion.

Relative Motion in One and Two Dimensions

Abstract:
By the end of this section, you will be able to:

* Explain the concept of reference frames.
* Write the position and velocity vector equations for relative motion.
* Draw the position and velocity vectors for relative motion.
* Analyze one-dimensional and two-dimensional relative motion problems using the position and velocity vector equations.

Reference Frames

To discuss relative motion in one or more dimensions, we first introduce the concept of reference frame**s**. When we say an object has a certain velocity, we must state it has a velocity with respect to a given reference frame. In most examples we have examined so far, this reference frame has been Earth. If you say a person is sitting in a train moving at 10 m/s east, then you imply the person on the train is moving relative to the surface of Earth at this velocity, and Earth is the reference frame. We can expand our view of the motion of the person on the train and say Earth is spinning in its orbit around the Sun, in which case the motion becomes more complicated. In this case, the solar system is the reference frame. In summary, all discussion of relative motion must define the reference frames involved. We now develop a method to refer to reference frames in relative motion. This method makes an approximation that breaks down near the speed of light, where the more accurate methods or special relativity are needed, but is extremely accurate for everyday speeds (Relativity).

Relative Motion in One Dimension

We introduce relative motion in one dimension first, because the velocity vectors simplify to having only two possible directions. Take the example of the person sitting in a train moving east. If we choose east as the positive direction and Earth as the reference frame, then we can write the velocity of the train with respect to the Earth as ${\stackrel{\to }{v}}_{\text{TE}}=10{\rule{0.2em}{0ex}}\text{m}\text{/}\text{s}{\rule{0.2em}{0ex}}\stackrel{^}{i}$ east, where the subscripts TE refer to train and Earth. Let’s now say the person gets up out of her seat and walks toward the back of the train at 2 m/s. This tells us she has a velocity relative to the reference frame of the train. Since the person is walking west, in the negative direction, we write her velocity with respect to the train as ${\stackrel{\to }{v}}_{\text{PT}}=-2{\rule{0.2em}{0ex}}\text{m}\text{/}\text{s}{\rule{0.2em}{0ex}}\stackrel{^}{i}\text{.}$ We can add the two velocity vectors to find the velocity of the person with respect to Earth. This relative velocity is written as

Relative Velocity in Two Dimensions

We can now apply these concepts to describing motion in two dimensions. Consider a particle *P* and reference frames *S* and ${S}^{\prime },$ as shown in [link]. The position of the origin of ${S}^{\prime }$ as measured in *S* is ${\stackrel{\to }{r}}_{{S}^{\prime }S},$ the position of *P* as measured in ${S}^{\prime }$ is ${\stackrel{\to }{r}}_{P{S}^{\prime }},$ and the position of *P* as measured in *S* is ${\stackrel{\to }{r}}_{PS}.$

Newton's Laws of Motion

Introduction

Forces

Abstract:
By the end of this section, you will be able to:

* Distinguish between kinematics and dynamics
* Understand the definition of force
* Identify simple free-body diagrams
* Define the SI unit of force, the newton
* Describe force as a vector

Working Definition of Force

Dynamics is the study of the forces that cause objects and systems to move. To understand this, we need a working definition of force. An intuitive definition of force—that is, a push or a pull—is a good place to start. We know that a push or a pull has both magnitude and direction (therefore, it is a vector quantity), so we can define force as the push or pull on an object with a specific magnitude and direction. Force can be represented by vectors or expressed as a multiple of a standard force.

Development of the Force Concept

A quantitative definition of force can be based on some standard force, just as distance is measured in units relative to a standard length. One possibility is to stretch a spring a certain fixed distance ([link]) and use the force it exerts to pull itself back to its relaxed shape—called a *restoring force*—as a standard. The magnitude of all other forces can be considered as multiples of this standard unit of force. Many other possibilities exist for standard forces. Some alternative definitions of force will be given later in this chapter.

Vector Notation for Force

As previously discussed, force is a vector; it has both magnitude and direction. The SI unit of force is called the newton (abbreviated N), and 1 N is the force needed to accelerate an object with a mass of 1 kg at a rate of $1{\rule{0.2em}{0ex}}{\text{m/s}}^{2}$: $1{\rule{0.2em}{0ex}}\text{N}=1{\rule{0.2em}{0ex}}\text{kg}·{\text{m/s}}^{2}.$ An easy way to remember the size of a newton is to imagine holding a small apple; it has a weight of about 1 N.

Newton's First Law

Abstract:
By the end of this section, you will be able to:

* Describe Newton's first law of motion
* Recognize friction as an external force
* Define inertia
* Identify inertial reference frames
* Calculate equilibrium for a system

Gravitation and Inertia

Regardless of the scale of an object, whether a molecule or a subatomic particle, two properties remain valid and thus of interest to physics: gravitation and inertia. Both are connected to mass. Roughly speaking, *mass* is a measure of the amount of matter in something. *Gravitation* is the attraction of one mass to another, such as the attraction between yourself and Earth that holds your feet to the floor. The magnitude of this attraction is your weight, and it is a force.

Inertial Reference Frames

Earlier, we stated Newton’s first law as “A body at rest remains at rest or, if in motion, remains in motion at constant velocity unless acted on by a net external force.” It can also be stated as “Every body remains in its state of uniform motion in a straight line unless it is compelled to change that state by forces acting on it.” To Newton, “uniform motion in a straight line” meant constant velocity, which includes the case of zero velocity, or rest. Therefore, the first law says that the velocity of an object remains constant if the net force on it is zero.

Newton’s First Law and Equilibrium

Newton’s first law tells us about the equilibrium of a system, which is the state in which the forces on the system are balanced. Returning to Forces and the ice skaters in [link], we know that the forces ${\stackrel{\to }{F}}_{1}$ and ${\stackrel{\to }{F}}_{2}$ combine to form a resultant force, or the net external force: ${\stackrel{\to }{F}}_{\text{R}}={\stackrel{\to }{F}}_{\text{net}}={\stackrel{\to }{F}}_{1}+{\stackrel{\to }{F}}_{2}.$ To create equilibrium, we require a balancing force that will produce a net force of zero. This force must be equal in magnitude but opposite in direction to ${\stackrel{\to }{F}}_{\text{R}},$ which means the vector must be $\text{−}{\stackrel{\to }{F}}_{\text{R}}.$ Referring to the ice skaters, for which we found ${\stackrel{\to }{F}}_{\text{R}}$ to be $30.0\stackrel{^}{i}+40.0\stackrel{^}{j}{\rule{0.2em}{0ex}}\text{N}$, we can determine the balancing force by simply finding $\text{−}{\stackrel{\to }{F}}_{\text{R}}=-30.0\stackrel{^}{i}-40.0\stackrel{^}{j}{\rule{0.2em}{0ex}}\text{N}.$ See the free-body diagram in [link](b).

Newton's Second Law

Abstract:
By the end of this section, you will be able to:

* Distinguish between external and internal forces
* Describe Newton's second law of motion
* Explain the dependence of acceleration on net force and mass

Force and Acceleration

First, what do we mean by a change in motion? The answer is that a change in motion is equivalent to a change in velocity. A change in velocity means, by definition, that there is acceleration. Newton’s first law says that a net external force causes a change in motion; thus, we see that a *net external force causes nonzero acceleration*.

Component Form of Newton’s Second Law

We have developed Newton’s second law and presented it as a vector equation in [link]. This vector equation can be written as three component equations:

Newton’s Second Law and Momentum

Newton actually stated his second law in terms of momentum: “The instantaneous rate at which a body’s momentum changes is equal to the net force acting on the body.” (“Instantaneous rate” implies that the derivative is involved.) This can be given by the vector equation

Mass and Weight

Abstract:
By the end of this section, you will be able to:

* Explain the difference between mass and weight
* Explain why objects falling through the air are never truly in free fall
* Describe the concept of weightlessness

Units of Force

The equation ${F}_{\text{net}}=ma$ is used to define net force in terms of mass, length, and time. As explained earlier, the SI unit of force is the newton. Since ${F}_{\text{net}}=ma,$

Weight and Gravitational Force

When an object is dropped, it accelerates toward the center of Earth. Newton’s second law says that a net force on an object is responsible for its acceleration. If air resistance is negligible, the net force on a falling object is the gravitational force, commonly called its weight $\stackrel{\to }{w}$, or its force due to gravity acting on an object of mass *m*. Weight can be denoted as a vector because it has a direction; *down* is, by definition, the direction of gravity, and hence, weight is a downward force. The magnitude of weight is denoted as *w*. Galileo was instrumental in showing that, in the absence of air resistance, all objects fall with the same acceleration *g*. Using Galileo’s result and Newton’s second law, we can derive an equation for weight.

Newton’s Third Law

Abstract:
By the end of this section, you will be able to:

* State Newton’s third law of motion
* Identify the action and reaction forces in different situations
* Apply Newton’s third law to define systems and solve problems of motion

Common Forces

Abstract:
By the end of this section, you will be able to:

* Define normal and tension forces
* Distinguish between real and fictitious forces
* Apply Newton’s laws of motion to solve problems involving a variety of forces

A Catalog of Forces: Normal, Tension, and Other Examples of Forces

A catalog of forces will be useful for reference as we solve various problems involving force and motion. These forces include normal force, tension, friction, and spring force.

Real Forces and Inertial Frames

There is another distinction among forces: Some forces are real, whereas others are not. *Real forces* have some physical origin, such as a gravitational pull. In contrast, *fictitious forces* arise simply because an observer is in an accelerating or noninertial frame of reference, such as one that rotates (like a merry-go-round) or undergoes linear acceleration (like a car slowing down). For example, if a satellite is heading due north above Earth’s Northern Hemisphere, then to an observer on Earth, it will appear to experience a force to the west that has no physical origin. Instead, Earth is rotating toward the east and moves east under the satellite. In Earth’s frame, this looks like a westward force on the satellite, or it can be interpreted as a violation of Newton’s first law (the law of inertia). We can identify a fictitious force by asking the question, “What is the reaction force?” If we cannot name the reaction force, then the force we are considering is fictitious. In the example of the satellite, the reaction force would have to be an eastward force on Earth. Recall that an inertial frame of reference is one in which all forces are real and, equivalently, one in which Newton’s laws have the simple forms given in this chapter.

Drawing Free-Body Diagrams

Abstract:
By the end of this section, you will be able to:

* Explain the rules for drawing a free-body diagram
* Construct free-body diagrams for different situations

Applications of Newton's Laws

Introduction

Solving Problems with Newton’s Laws

Abstract:
By the end of this section, you will be able to:

* Apply problem-solving techniques to solve for quantities in more complex systems of forces
* Use concepts from kinematics to solve problems using Newton’s laws of motion
* Solve more complex equilibrium problems
* Solve more complex acceleration problems
* Apply calculus to more advanced dynamics problems

Problem-Solving Strategies

We follow here the basics of problem solving presented earlier in this text, but we emphasize specific strategies that are useful in applying Newton’s laws of motion. Once you identify the physical principles involved in the problem and determine that they include Newton’s laws of motion, you can apply these steps to find a solution. These techniques also reinforce concepts that are useful in many other areas of physics. Many problem-solving strategies are stated outright in the worked examples, so the following techniques should reinforce skills you have already begun to develop.

Particle Equilibrium

Recall that a particle in equilibrium is one for which the external forces are balanced. Static equilibrium involves objects at rest, and dynamic equilibrium involves objects in motion without acceleration, but it is important to remember that these conditions are relative. For example, an object may be at rest when viewed from our frame of reference, but the same object would appear to be in motion when viewed by someone moving at a constant velocity. We now make use of the knowledge attained in Newton’s Laws of Motion, regarding the different types of forces and the use of free-body diagrams, to solve additional problems in particle equilibrium.

Particle Acceleration

We have given a variety of examples of particles in equilibrium. We now turn our attention to particle acceleration problems, which are the result of a nonzero net force. Refer again to the steps given at the beginning of this section, and notice how they are applied to the following examples.

Newton’s Laws of Motion and Kinematics

Physics is most interesting and most powerful when applied to general situations that involve more than a narrow set of physical principles. Newton’s laws of motion can also be integrated with other concepts that have been discussed previously in this text to solve problems of motion. For example, forces produce accelerations, a topic of kinematics, and hence the relevance of earlier chapters.

Friction

Abstract:
By the end of this section, you will be able to:

* Describe the general characteristics of friction
* List the various types of friction
* Calculate the magnitude of static and kinetic friction, and use these in problems involving Newton’s laws of motion

Static and Kinetic Friction

The basic definition of friction is relatively simple to state.

Friction and the Inclined Plane

One situation where friction plays an obvious role is that of an object on a slope. It might be a crate being pushed up a ramp to a loading dock or a skateboarder coasting down a mountain, but the basic physics is the same. We usually generalize the sloping surface and call it an inclined plane but then pretend that the surface is flat. Let’s look at an example of analyzing motion on an inclined plane with friction.

Atomic-Scale Explanations of Friction

The simpler aspects of friction dealt with so far are its macroscopic (large-scale) characteristics. Great strides have been made in the atomic-scale explanation of friction during the past several decades. Researchers are finding that the atomic nature of friction seems to have several fundamental characteristics. These characteristics not only explain some of the simpler aspects of friction—they also hold the potential for the development of nearly friction-free environments that could save hundreds of billions of dollars in energy which is currently being converted (unnecessarily) into heat.

Centripetal Force

Abstract:
By the end of this section, you will be able to:

* Explain the equation for centripetal acceleration
* Apply Newton’s second law to develop the equation for centripetal force
* Use circular motion concepts in solving problems involving Newton’s laws of motion

Banked Curves

Let us now consider banked curve**s**, where the slope of the road helps you negotiate the curve ([link]). The greater the angle $\theta$, the faster you can take the curve. Race tracks for bikes as well as cars, for example, often have steeply banked curves. In an “ideally banked curve,” the angle $\theta$ is such that you can negotiate the curve at a certain speed without the aid of friction between the tires and the road. We will derive an expression for $\theta$ for an ideally banked curve and consider an example related to it.

Inertial Forces and Noninertial (Accelerated) Frames: The Coriolis Force

What do taking off in a jet airplane, turning a corner in a car, riding a merry-go-round, and the circular motion of a tropical cyclone have in common? Each exhibits inertial forces—fictitious forces that merely seem to arise from motion, because the observer’s frame of reference is accelerating or rotating. When taking off in a jet, most people would agree it feels as if you are being pushed back into the seat as the airplane accelerates down the runway. Yet a physicist would say that *you* tend to remain stationary while the *seat* pushes forward on you. An even more common experience occurs when you make a tight curve in your car—say, to the right ([link]). You feel as if you are thrown (that is, *forced*) toward the left relative to the car. Again, a physicist would say that *you* are going in a straight line (recall Newton’s first law) but the *car* moves to the right, not that you are experiencing a force from the left.

Drag Force and Terminal Speed

Abstract:
By the end of this section, you will be able to:

* Express the drag force mathematically
* Describe applications of the drag force
* Define terminal velocity
* Determine an object’s terminal velocity given its mass

Drag Forces

Like friction, the drag force always opposes the motion of an object. Unlike simple friction, the drag force is proportional to some function of the velocity of the object in that fluid. This functionality is complicated and depends upon the shape of the object, its size, its velocity, and the fluid it is in. For most large objects such as cyclists, cars, and baseballs not moving too slowly, the magnitude of the drag force ${F}_{\text{D}}$ is proportional to the square of the speed of the object. We can write this relationship mathematically as ${F}_{\text{D}}\propto {v}^{2}.$ When taking into account other factors, this relationship becomes

Terminal Velocity

Some interesting situations connected to Newton’s second law occur when considering the effects of drag forces upon a moving object. For instance, consider a skydiver falling through air under the influence of gravity. The two forces acting on him are the force of gravity and the drag force (ignoring the small buoyant force). The downward force of gravity remains constant regardless of the velocity at which the person is moving. However, as the person’s velocity increases, the magnitude of the drag force increases until the magnitude of the drag force is equal to the gravitational force, thus producing a net force of zero. A zero net force means that there is no acceleration, as shown by Newton’s second law. At this point, the person’s velocity remains constant and we say that the person has reached his terminal velocity $\left({v}_{\text{T}}\right).$ Since ${F}_{\text{D}}$ is proportional to the speed squared, a heavier skydiver must go faster for ${F}_{\text{D}}$ to equal his weight. Let’s see how this works out more quantitatively.

The Calculus of Velocity-Dependent Frictional Forces

When a body slides across a surface, the frictional force on it is approximately constant and given by ${\mu }_{\text{k}}N.$ Unfortunately, the frictional force on a body moving through a liquid or a gas does not behave so simply. This drag force is generally a complicated function of the body’s velocity. However, for a body moving in a straight line at moderate speeds through a liquid such as water, the frictional force can often be approximated by

Work and Kinetic Energy

Introduction

Work

Abstract:
By the end of this section, you will be able to:

* Represent the work done by any force
* Evaluate the work done for various forces

Work Done by Constant Forces and Contact Forces

The simplest work to evaluate is that done by a force that is constant in magnitude and direction. In this case, we can factor out the force; the remaining integral is just the total displacement, which only depends on the end points *A* and *B*, but not on the path between them:

Work Done by Forces that Vary

In general, forces may vary in magnitude and direction at points in space, and paths between two points may be curved. The infinitesimal work done by a variable force can be expressed in terms of the components of the force and the displacement along the path,

Kinetic Energy

Abstract:
By the end of this section, you will be able to:

* Calculate the kinetic energy of a particle given its mass and its velocity or momentum
* Evaluate the kinetic energy of a body, relative to different frames of reference

Work-Energy Theorem

Abstract:
By the end of this section, you will be able to:

* Apply the work-energy theorem to find information about the motion of a particle, given the forces acting on it
* Use the work-energy theorem to find information about the forces acting on a particle, given information about its motion

Power

Abstract:
By the end of this section, you will be able to:

* Relate the work done during a time interval to the power delivered
* Find the power expended by a force acting on a moving body

Potential Energy and Conservation of Energy

Introduction

Potential Energy of a System

Abstract:
By the end of this section, you will be able to:

* Relate the difference of potential energy to work done on a particle for a system without friction or air drag
* Explain the meaning of the zero of the potential energy function for a system
* Calculate and apply the gravitational potential energy for an object near Earth’s surface and the elastic potential energy of a mass-spring system

Potential Energy Basics

In Motion in Two and Three Dimensions, we analyzed the motion of a projectile, like kicking a football in [link]. For this example, let’s ignore friction and air resistance and begin the analysis immediately after the kick, at the start of the projectile motion of the ball. As the football rises, the work done by the gravitational force on the football is negative, because the ball’s displacement is positive vertically and the force due to gravity is negative vertically. We also noted that the ball slowed down until it reached its highest point in the motion, thereby decreasing the ball’s kinetic energy. This loss in kinetic energy translates to a gain in gravitational potential energy of the football-Earth system.

Systems of Several Particles

In general, a system of interest could consist of several particles. The difference in the potential energy of the system is the negative of the work done by gravitational or elastic forces, which, as we will see in the next section, are conservative forces. The potential energy difference depends only on the initial and final positions of the particles, and on some parameters that characterize the interaction (like mass for gravity or the spring constant for a Hooke’s law force).

Types of Potential Energy

For each type of interaction present in a system, you can label a corresponding type of potential energy. The total potential energy of the system is the sum of the potential energies of all the types. (This follows from the additive property of the dot product in the expression for the work done.) Let’s look at some specific examples of types of potential energy discussed in Work. First, we consider each of these forces when acting separately, and then when both act together.

Conservative and Non-Conservative Forces

Abstract:
By the end of this section, you will be able to:

* Characterize a conservative force in several different ways
* Specify mathematical conditions that must be satisfied by a conservative force and its components
* Relate the conservative force between particles of a system to the potential energy of the system
* Calculate the components of a conservative force in various cases

Conservation of Energy

Abstract:
By the end of this section, you will be able to:

* Formulate the principle of conservation of mechanical energy, with or without the presence of non-conservative forces
* Use the conservation of mechanical energy to calculate various properties of simple systems

Systems with a Single Particle or Object

We first consider a system with a single particle or object. Returning to our development of [link], recall that we first separated all the forces acting on a particle into conservative and non-conservative types, and wrote the work done by each type of force as a separate term in the work-energy theorem. We then replaced the work done by the conservative forces by the change in the potential energy of the particle, combining it with the change in the particle’s kinetic energy to get [link]. Now, we write this equation without the middle step and define the sum of the kinetic and potential energies, $K+U=E;$ to be the mechanical energy of the particle.

Systems with Several Particles or Objects

Systems generally consist of more than one particle or object. However, the conservation of mechanical energy, in one of the forms in [link] or [link], is a fundamental law of physics and applies to any system. You just have to include the kinetic and potential energies of all the particles, and the work done by all the non-conservative forces acting on them. Until you learn more about the dynamics of systems composed of many particles, in Linear Momentum and Collisions, Fixed-Axis Rotation, and Angular Momentum, it is better to postpone discussing the application of energy conservation to then.

Potential Energy Diagrams and Stability

Abstract:
By the end of this section, you will be able to:

* Create and interpret graphs of potential energy
* Explain the connection between stability and potential energy

Sources of Energy

Abstract:
By the end of this section, you will be able to:

* Describe energy transformations and conversions in general terms
* Explain what it means for an energy source to be renewable or nonrenewable

Linear Momentum and Collisions

Introduction

Linear Momentum

Abstract:
By the end of this section, you will be able to:

* Explain what momentum is, physically
* Calculate the momentum of a moving object

Impulse and Collisions

Abstract:
By the end of this section, you will be able to:

* Explain what an impulse is, physically
* Describe what an impulse does
* Relate impulses to collisions
* Apply the impulse-momentum theorem to solve problems

Effect of Impulse

Since an impulse is a force acting for some amount of time, it causes an object’s motion to change. Recall [link]:

Momentum and Force

In [link], we obtained an important relationship:

Conservation of Linear Momentum

Abstract:
By the end of this section, you will be able to:

* Explain the meaning of “conservation of momentum”
* Correctly identify if a system is, or is not, closed
* Define a system whose momentum is conserved
* Mathematically express conservation of momentum for a given system
* Calculate an unknown quantity using conservation of momentum

Requirements for Momentum Conservation

There is a complication, however. A system must meet two requirements for its momentum to be conserved:

The Meaning of ‘System’

A system (mechanical) is the collection of objects in whose motion (kinematics and dynamics) you are interested. If you are analyzing the bounce of a ball on the ground, you are probably only interested in the motion of the ball, and not of Earth; thus, the ball is your system. If you are analyzing a car crash, the two cars together compose your system ([link]).

Types of Collisions

Abstract:
By the end of this section, you will be able to:

* Identify the type of collision
* Correctly label a collision as elastic or inelastic
* Use kinetic energy along with momentum and impulse to analyze a collision

Explosions

The first possibility is that a single object may break apart into two or more pieces. An example of this is a firecracker, or a bow and arrow, or a rocket rising through the air toward space. These can be difficult to analyze if the number of fragments after the collision is more than about three or four; but nevertheless, the total momentum of the system before and after the explosion is identical.

Inelastic

The second possibility is the reverse: that two or more objects collide with each other and stick together, thus (after the collision) forming one single composite object. The total mass of this composite object is the sum of the masses of the original objects, and the new single object moves with a velocity dictated by the conservation of momentum. However, it turns out again that, although the total momentum of the system of objects remains constant, the kinetic energy doesn’t; but this time, the kinetic energy decreases. This type of collision is called inelastic.

Elastic

The extreme case on the other end is if two or more objects approach each other, collide, and bounce off each other, moving away from each other at the same relative speed at which they approached each other. In this case, the total kinetic energy of the system is conserved. Such an interaction is called elastic.

Subatomic Collisions and Momentum

Conservation of momentum is crucial to our understanding of atomic and subatomic particles because much of what we know about these particles comes from collision experiments.

Collisions in Multiple Dimensions

Abstract:
By the end of this section, you will be able to:

* Express momentum as a two-dimensional vector
* Write equations for momentum conservation in component form
* Calculate momentum in two dimensions, as a vector quantity

Center of Mass

Abstract:
By the end of this section, you will be able to:

* Explain the meaning and usefulness of the concept of center of mass
* Calculate the center of mass of a given system
* Apply the center of mass concept in two and three dimensions
* Calculate the velocity and acceleration of the center of mass

Internal and External Forces

Suppose we have an extended object of mass *M*, made of *N* interacting particles. Let’s label their masses as ${m}_{j}$, where $j=1,2,3,\text{…},N$. Note that

Force and Momentum

Remember that our actual goal is to determine the equation of motion for the entire object (the entire system of particles). To that end, let’s define:

Center of Mass

Our next task is to determine what part of the extended object, if any, is obeying [link].

Center of Mass of Continuous Objects

If the object in question has its mass distributed uniformly in space, rather than as a collection of discrete particles, then ${m}_{j}\to dm$, and the summation becomes an integral:

Center of Mass and Conservation of Momentum

How does all this connect to conservation of momentum?

Rocket Propulsion

Abstract:
By the end of this section, you will be able to:

* Describe the application of conservation of momentum when the mass changes with time, as well as the velocity
* Calculate the speed of a rocket in empty space, at some time, given initial conditions
* Calculate the speed of a rocket in Earth’s gravity field, at some time, given initial conditions

Physical Analysis

Here’s a description of what happens, so that you get a feel for the physics involved.

Rocket in a Gravitational Field

Let’s now analyze the velocity change of the rocket during the launch phase, from the surface of Earth. To keep the math manageable, we’ll restrict our attention to distances for which the acceleration caused by gravity can be treated as a constant *g*.

Fixed-Axis Rotation

Introduction

Rotational Variables

Abstract:
By the end of this section, you will be able to:

* Describe the physical meaning of rotational variables as applied to fixed-axis rotation
* Explain how angular velocity is related to tangential speed
* Calculate the instantaneous angular velocity given the angular position function
* Find the angular velocity and angular acceleration in a rotating system
* Calculate the average angular acceleration when the angular velocity is changing
* Calculate the instantaneous angular acceleration given the angular velocity function

Angular Velocity

Uniform circular motion (discussed previously in Motion in Two and Three Dimensions) is motion in a circle at constant speed. Although this is the simplest case of rotational motion, it is very useful for many situations, and we use it here to introduce rotational variables.

Angular Acceleration

We have just discussed angular velocity for uniform circular motion, but not all motion is uniform. Envision an ice skater spinning with his arms outstretched—when he pulls his arms inward, his angular velocity increases. Or think about a computer’s hard disk slowing to a halt as the angular velocity decreases. We will explore these situations later, but we can already see a need to define an angular acceleration for describing situations where $\omega$ changes. The faster the change in $\omega$, the greater the angular acceleration. We define the instantaneous angular acceleration $\alpha$ as the derivative of angular velocity with respect to time:

Rotation with Constant Angular Acceleration

Abstract:
By the end of this section, you will be able to:

* Derive the kinematic equations for rotational motion with constant angular acceleration
* Select from the kinematic equations for rotational motion with constant angular acceleration the appropriate equations to solve for unknowns in the analysis of systems undergoing fixed-axis rotation
* Use solutions found with the kinematic equations to verify the graphical analysis of fixed-axis rotation with constant angular acceleration

Kinematics of Rotational Motion

Using our intuition, we can begin to see how the rotational quantities $\theta ,$ $\omega ,$ $\alpha$, and *t* are related to one another. For example, we saw in the preceding section that if a flywheel has an angular acceleration in the same direction as its angular velocity vector, its angular velocity increases with time and its angular displacement also increases. On the contrary, if the angular acceleration is opposite to the angular velocity vector, its angular velocity decreases with time. We can describe these physical situations and many others with a consistent set of rotational kinematic equations under a constant angular acceleration. The method to investigate rotational motion in this way is called kinematics of rotational motion.

Applying the Equations for Rotational Motion

Now we can apply the key kinematic relations for rotational motion to some simple examples to get a feel for how the equations can be applied to everyday situations.

Relating Angular and Translational Quantities

Abstract:
By the end of this section, you will be able to:

* Given the linear kinematic equation, write the corresponding rotational kinematic equation
* Calculate the linear distances, velocities, and accelerations of points on a rotating system given the angular velocities and accelerations

Angular vs. Linear Variables

In Rotational Variables, we introduced angular variables. If we compare the rotational definitions with the definitions of linear kinematic variables from Motion Along a Straight Line and Motion in Two and Three Dimensions, we find that there is a mapping of the linear variables to the rotational ones. Linear position, velocity, and acceleration have their rotational counterparts, as we can see when we write them side by side:

Relationships between Rotational and Translational Motion

We can look at two relationships between rotational and translational motion.

Moment of Inertia and Rotational Kinetic Energy

Abstract:
By the end of this section, you will be able to:

* Describe the differences between rotational and translational kinetic energy
* Define the physical concept of moment of inertia in terms of the mass distribution from the rotational axis
* Explain how the moment of inertia of rigid bodies affects their rotational kinetic energy
* Use conservation of mechanical energy to analyze systems undergoing both rotation and translation
* Calculate the angular velocity of a rotating system when there are energy losses due to nonconservative forces

Rotational Kinetic Energy

Any moving object has kinetic energy. We know how to calculate this for a body undergoing translational motion, but how about for a rigid body undergoing rotation? This might seem complicated because each point on the rigid body has a different velocity. However, we can make use of angular velocity—which is the same for the entire rigid body—to express the kinetic energy for a rotating object. [link] shows an example of a very energetic rotating body: an electric grindstone propelled by a motor. Sparks are flying, and noise and vibration are generated as the grindstone does its work. This system has considerable energy, some of it in the form of heat, light, sound, and vibration. However, most of this energy is in the form of rotational kinetic energy.

Moment of Inertia

If we compare [link] to the way we wrote kinetic energy in Work and Kinetic Energy, $\left(\frac{1}{2}m{v}^{2}\right)$, this suggests we have a new rotational variable to add to our list of our relations between rotational and translational variables. The quantity $\sum _{j}{m}_{j}{r}_{j}^{2}$ is the counterpart for mass in the equation for rotational kinetic energy. This is an important new term for rotational motion. This quantity is called the moment of inertia *I*, with units of $\text{kg}·{\text{m}}^{2}$:

Applying Rotational Kinetic Energy

Now let’s apply the ideas of rotational kinetic energy and the moment of inertia table to get a feeling for the energy associated with a few rotating objects. The following examples will also help get you comfortable using these equations. First, let’s look at a general problem-solving strategy for rotational energy.

Calculating Moments of Inertia

Abstract:
By the end of this section, you will be able to:

* Calculate the moment of inertia for uniformly shaped, rigid bodies
* Apply the parallel axis theorem to find the moment of inertia about any axis parallel to one already known
* Calculate the moment of inertia for compound objects

Moment of Inertia

We defined the moment of inertia *I* of an object to be $I=\sum _{i}{m}_{i}{r}_{i}^{2}$ for all the point masses that make up the object. Because *r* is the distance to the axis of rotation from each piece of mass that makes up the object, the moment of inertia for any object depends on the chosen axis. To see this, let’s take a simple example of two masses at the end of a massless (negligibly small mass) rod ([link]) and calculate the moment of inertia about two different axes. In this case, the summation over the masses is simple because the two masses at the end of the barbell can be approximated as point masses, and the sum therefore has only two terms.

The Parallel-Axis Theorem

The similarity between the process of finding the moment of inertia of a rod about an axis through its middle and about an axis through its end is striking, and suggests that there might be a simpler method for determining the moment of inertia for a rod about any axis parallel to the axis through the center of mass. Such an axis is called a parallel axis. There is a theorem for this, called the parallel-axis theorem, which we state here but do not derive in this text.

Torque

Abstract:
By the end of this section, you will be able to:

* Describe how the magnitude of a torque depends on the magnitude of the lever arm and the angle the force vector makes with the lever arm
* Determine the sign (positive or negative) of a torque using the right-hand rule
* Calculate individual torques about a common axis and sum them to find the net torque

Defining Torque

So far we have defined many variables that are rotational equivalents to their translational counterparts. Let’s consider what the counterpart to force must be. Since forces change the translational motion of objects, the rotational counterpart must be related to changing the rotational motion of an object about an axis. We call this rotational counterpart torque.

Calculating Net Torque for Rigid Bodies on a Fixed Axis

In the following examples, we calculate the torque both abstractly and as applied to a rigid body.

Newton’s Second Law for Rotation

Abstract:
By the end of this section, you will be able to:

* Calculate the torques on rotating systems about a fixed axis to find the angular acceleration
* Explain how changes in the moment of inertia of a rotating system affect angular acceleration with a fixed applied torque

Newton’s Second Law for Rotation

We have thus far found many counterparts to the translational terms used throughout this text, most recently, torque, the rotational analog to force. This raises the question: Is there an analogous equation to Newton’s second law, $\Sigma \stackrel{\to }{F}=m\stackrel{\to }{a},$ which involves torque and rotational motion? To investigate this, we start with Newton’s second law for a single particle rotating around an axis and executing circular motion. Let’s exert a force $\stackrel{\to }{F}$ on a point mass *m* that is at a distance *r* from a pivot point ([link]). The particle is constrained to move in a circular path with fixed radius and the force is tangent to the circle. We apply Newton’s second law to determine the magnitude of the acceleration *$a=F\text{/}m$* in the direction of $\stackrel{\to }{F}$. Recall that the magnitude of the tangential acceleration is proportional to the magnitude of the angular acceleration by $a=r\alpha$. Substituting this expression into Newton’s second law, we obtain

Deriving Newton’s Second Law for Rotation in Vector Form

As before, when we found the angular acceleration, we may also find the torque vector. The second law $\Sigma \stackrel{\to }{F}=m\stackrel{\to }{a}$ tells us the relationship between net force and how to change the translational motion of an object. We have a vector rotational equivalent of this equation, which can be found by using [link] and [link]. [link] relates the angular acceleration to the position and tangential acceleration vectors:

Applying the Rotational Dynamics Equation

Before we apply the rotational dynamics equation to some everyday situations, let’s review a general problem-solving strategy for use with this category of problems.

Work and Power for Rotational Motion

Abstract:
By the end of this section, you will be able to:

* Use the work-energy theorem to analyze rotation to find the work done on a system when it is rotated about a fixed axis for a finite angular displacement
* Solve for the angular velocity of a rotating rigid body using the work-energy theorem
* Find the power delivered to a rotating rigid body given the applied torque and angular velocity
* Summarize the rotational variables and equations and relate them to their translational counterparts

Work for Rotational Motion

Now that we have determined how to calculate kinetic energy for rotating rigid bodies, we can proceed with a discussion of the work done on a rigid body rotating about a fixed axis. [link] shows a rigid body that has rotated through an angle $d\theta$ from *A* to *B* while under the influence of a force $\stackrel{\to }{F}$. The external force $\stackrel{\to }{F}$ is applied to point *P*, whose position is $\stackrel{\to }{r}$, and the rigid body is constrained to rotate about a fixed axis that is perpendicular to the page and passes through *O*. The rotational axis is fixed, so the vector $\stackrel{\to }{r}$ moves in a circle of radius *r*, and the vector $d\stackrel{\to }{s}$ is perpendicular to $\stackrel{\to }{r}.$

Power for Rotational Motion

Power always comes up in the discussion of applications in engineering and physics. Power for rotational motion is equally as important as power in linear motion and can be derived in a similar way as in linear motion when the force is a constant. The linear power when the force is a constant is $P=\stackrel{\to }{F}·\stackrel{\to }{v}$. If the net torque is constant over the angular displacement, [link] simplifies and the net torque can be taken out of the integral. In the following discussion, we assume the net torque is constant. We can apply the definition of power derived in Power to rotational motion. From Work and Kinetic Energy, the instantaneous power (or just power) is defined as the rate of doing work,

Rotational and Translational Relationships Summarized

The rotational quantities and their linear analog are summarized in three tables. [link] summarizes the rotational variables for circular motion about a fixed axis with their linear analogs and the connecting equation, except for the centripetal acceleration, which stands by itself. [link] summarizes the rotational and translational kinematic equations. [link] summarizes the rotational dynamics equations with their linear analogs.

Angular Momentum

Introduction

Rolling Motion

Abstract:
By the end of this section, you will be able to:

* Describe the physics of rolling motion without slipping
* Explain how linear variables are related to angular variables for the case of rolling motion without slipping
* Find the linear and angular accelerations in rolling motion with and without slipping
* Calculate the static friction force associated with rolling motion without slipping
* Use energy conservation to analyze rolling motion

Rolling Motion without Slipping

People have observed rolling motion without slipping ever since the invention of the wheel. For example, we can look at the interaction of a car’s tires and the surface of the road. If the driver depresses the accelerator to the floor, such that the tires spin without the car moving forward, there must be kinetic friction between the wheels and the surface of the road. If the driver depresses the accelerator slowly, causing the car to move forward, then the tires roll without slipping. It is surprising to most people that, in fact, the bottom of the wheel is at rest with respect to the ground, indicating there must be static friction between the tires and the road surface. In [link], the bicycle is in motion with the rider staying upright. The tires have contact with the road surface, and, even though they are rolling, the bottoms of the tires deform slightly, do not slip, and are at rest with respect to the road surface for a measurable amount of time. There must be static friction between the tire and the road surface for this to be so.

Rolling Motion with Slipping

In the case of rolling motion with slipping, we must use the coefficient of kinetic friction, which gives rise to the kinetic friction force since static friction is not present. The situation is shown in [link]. In the case of slipping, ${v}_{\text{CM}}-R\omega \ne 0$, because point *P* on the wheel is not at rest on the surface, and ${v}_{P}\ne 0$. Thus, $\omega \ne \frac{{v}_{\text{CM}}}{R},\alpha \ne \frac{{a}_{\text{CM}}}{R}$.

Conservation of Mechanical Energy in Rolling Motion

In the preceding chapter, we introduced rotational kinetic energy. Any rolling object carries rotational kinetic energy, as well as translational kinetic energy and potential energy if the system requires. Including the gravitational potential energy, the total mechanical energy of an object rolling is

Angular Momentum

Abstract:
By the end of this section, you will be able to:

* Describe the vector nature of angular momentum
* Find the total angular momentum and torque about a designated origin of a system of particles
* Calculate the angular momentum of a rigid body rotating about a fixed axis
* Calculate the torque on a rigid body rotating about a fixed axis
* Use conservation of angular momentum in the analysis of objects that change their rotation rate

Angular Momentum of a Single Particle

[link] shows a particle at a position $\stackrel{\to }{r}$ with linear momentum $\stackrel{\to }{p}=m\stackrel{\to }{v}$ with respect to the origin. Even if the particle is not rotating about the origin, we can still define an angular momentum in terms of the position vector and the linear momentum.

Angular Momentum of a System of Particles

The angular momentum of a system of particles is important in many scientific disciplines, one being astronomy. Consider a spiral galaxy, a rotating island of stars like our own Milky Way. The individual stars can be treated as point particles, each of which has its own angular momentum. The vector sum of the individual angular momenta give the total angular momentum of the galaxy. In this section, we develop the tools with which we can calculate the total angular momentum of a system of particles.

Angular Momentum of a Rigid Body

We have investigated the angular momentum of a single particle, which we generalized to a system of particles. Now we can use the principles discussed in the previous section to develop the concept of the angular momentum of a rigid body. Celestial objects such as planets have angular momentum due to their spin and orbits around stars. In engineering, anything that rotates about an axis carries angular momentum, such as flywheels, propellers, and rotating parts in engines. Knowledge of the angular momenta of these objects is crucial to the design of the system in which they are a part.

Conservation of Angular Momentum

Abstract:
By the end of this section, you will be able to:

* Apply conservation of angular momentum to determine the angular velocity of a rotating system in which the moment of inertia is changing
* Explain how the rotational kinetic energy changes when a system undergoes changes in both moment of inertia and angular velocity

Precession of a Gyroscope

Abstract:
By the end of this section, you will be able to:

* Describe the physical processes underlying the phenomenon of precession
* Calculate the precessional angular velocity of a gyroscope

Static Equilibrium and Elasticity

Introduction

Conditions for Static Equilibrium

Abstract:
By the end of this section, you will be able to:

* Identify the physical conditions of static equilibrium.
* Draw a free-body diagram for a rigid body acted on by forces.
* Explain how the conditions for equilibrium allow us to solve statics problems.

Examples of Static Equilibrium

Abstract:
By the end of this section, you will be able to:

* Identify and analyze static equilibrium situations
* Set up a free-body diagram for an extended object in static equilibrium
* Set up and solve static equilibrium conditions for objects in equilibrium in various physical situations

Stress, Strain, and Elastic Modulus

Abstract:
By the end of this section, you will be able to:

* Explain the concepts of stress and strain in describing elastic deformations of materials
* Describe the types of elastic deformation of objects and materials

Tensile or Compressive Stress, Strain, and Young’s Modulus

Tension or compression occurs when two antiparallel forces of equal magnitude act on an object along only one of its dimensions, in such a way that the object does not move. One way to envision such a situation is illustrated in [link]. A rod segment is either stretched or squeezed by a pair of forces acting along its length and perpendicular to its cross-section. The net effect of such forces is that the rod changes its length from the original length ${L}_{0}$ that it had before the forces appeared, to a new length *L* that it has under the action of the forces. This change in length $\text{Δ}L=L-{L}_{0}$ may be either elongation (when *L* is larger than the original length ${L}_{0}\right)$ or contraction (when *L* is smaller than the original length ${L}_{0}\right).$ Tensile stress and strain occur when the forces are stretching an object, causing its elongation, and the length change $\text{Δ}L$ is positive. Compressive stress and strain occur when the forces are contracting an object, causing its shortening, and the length change $\text{Δ}L$ is negative.

Bulk Stress, Strain, and Modulus

When you dive into water, you feel a force pressing on every part of your body from all directions. What you are experiencing then is bulk stress, or in other words, pressure. Bulk stress always tends to decrease the volume enclosed by the surface of a submerged object. The forces of this “squeezing” are always perpendicular to the submerged surface [link]. The effect of these forces is to decrease the volume of the submerged object by an amount $\text{Δ}V$ compared with the volume ${V}_{0}$ of the object in the absence of bulk stress. This kind of deformation is called bulk strain and is described by a change in volume relative to the original volume:

Shear Stress, Strain, and Modulus

The concepts of shear stress and strain concern only solid objects or materials. Buildings and tectonic plates are examples of objects that may be subjected to shear stresses. In general, these concepts do not apply to fluids.

Elasticity and Plasticity

Abstract:
By the end of this section, you will be able to:

* Explain the limit where a deformation of material is elastic
* Describe the range where materials show plastic behavior
* Analyze elasticity and plasticity on a stress-strain diagram

Gravitation

Introduction

Newton's Law of Universal Gravitation

Abstract:
By the end of this section, you will be able to:

* List the significant milestones in the history of gravitation
* Calculate the gravitational force between two point masses
* Estimate the gravitational force between collections of mass

The History of Gravitation

The earliest philosophers wondered why objects naturally tend to fall toward the ground. Aristotle (384–322 BCE) believed that it was the nature of rocks to seek Earth and the nature of fire to seek the Heavens. Brahmagupta (598~665 CE) postulated that Earth was a sphere and that objects possessed a natural affinity for it, falling toward the center from wherever they were located.

Newton’s Law of Universal Gravitation

Newton noted that objects at Earth’s surface (hence at a distance of ${R}_{\text{E}}$ from the center of Earth) have an acceleration of *g*, but the Moon, at a distance of about $60{\rule{0.2em}{0ex}}{R}_{\text{E}}$, has a centripetal acceleration about ${\left(60\right)}^{2}$ times smaller than *g*. He could explain this by postulating that a force exists between any two objects, whose magnitude is given by the product of the two masses divided by the square of the distance between them. We now know that this inverse square law is ubiquitous in nature, a function of geometry for point sources. The strength of any source at a distance *r* is spread over the surface of a sphere centered about the mass. The surface area of that sphere is proportional to ${r}^{2}$. In later chapters, we see this same form in the electromagnetic force.

The Cavendish Experiment

A century after Newton published his law of universal gravitation, Henry Cavendish determined the proportionality constant *G* by performing a painstaking experiment. He constructed a device similar to that shown in [link], in which small masses are suspended from a wire. Once in equilibrium, two fixed, larger masses are placed symmetrically near the smaller ones. The gravitational attraction creates a torsion (twisting) in the supporting wire that can be measured.

Gravitation Near Earth's Surface

Abstract:
By the end of this section, you will be able to:

* Explain the connection between the constants $G$ and $g$
* Determine the mass of an astronomical body from free-fall acceleration at its surface
* Describe how the value of $g$ varies due to location and Earth’s rotation

Weight

Recall that the acceleration of a free-falling object near Earth’s surface is approximately $g=9.80{\rule{0.2em}{0ex}}{\text{m/s}}^{2}$. The force causing this acceleration is called the weight of the object, and from Newton’s second law, it has the value *mg*. This weight is present regardless of whether the object is in free fall. We now know that this force is the gravitational force between the object and Earth. If we substitute *mg* for the magnitude of ${\stackrel{\to }{F}}_{12}$ in Newton’s law of universal gravitation, *m* for ${m}_{1}$, and ${M}_{\text{E}}$ for ${m}_{2}$, we obtain the scalar equation

The Gravitational Field

[link] is a scalar equation, giving the magnitude of the gravitational acceleration as a function of the distance from the center of the mass that causes the acceleration. But we could have retained the vector form for the force of gravity in [link], and written the acceleration in vector form as

Apparent Weight: Accounting for Earth’s Rotation

As we saw in Applications of Newton’s Laws, objects moving at constant speed in a circle have a centripetal acceleration directed toward the center of the circle, which means that there must be a net force directed toward the center of that circle. Since all objects on the surface of Earth move through a circle every 24 hours, there must be a net centripetal force on each object directed toward the center of that circle.

Results Away from the Equator

At the poles, ${a}_{\text{c}}\to 0$ and ${F}_{\text{s}}=mg$, just as is the case without rotation. At any other latitude $\lambda$, the situation is more complicated. The centripetal acceleration is directed toward point *P* in the figure, and the radius becomes $r={R}_{\text{E}}\text{cos}\lambda$. The *vector* sum of the weight and ${\stackrel{\to }{F}}_{\text{s}}$ must point toward point *P*, hence ${\stackrel{\to }{F}}_{\text{s}}$ no longer points away from the center of Earth. (The difference is small and exaggerated in the figure.) A plumb bob will always point along this deviated direction. All buildings are built aligned along this deviated direction, not along a radius through the center of Earth. For the tallest buildings, this represents a deviation of a few feet at the top.

Gravity Away from the Surface

Earlier we stated without proof that the law of gravitation applies to spherically symmetrical objects, where the mass of each body acts as if it were at the center of the body. Since [link] is derived from [link], it is also valid for symmetrical mass distributions, but both equations are valid only for values of $r\ge {R}_{\text{E}}$. As we saw in [link], at 400 km above Earth’s surface, where the International Space Station orbits, the value of *g* is $8.67{\rule{0.2em}{0ex}}{\text{m/s}}^{2}$. (We will see later that this is also the centripetal acceleration of the ISS.)

Gravitational Potential Energy and Total Energy

Abstract:
By the end of this section, you will be able to:

* Determine changes in gravitational potential energy over great distances
* Apply conservation of energy to determine escape velocity
* Determine whether astronomical bodies are gravitationally bound

Gravitational Potential Energy beyond Earth

We defined work and potential energy in Work and Kinetic Energy and Potential Energy and Conservation of Energy. The usefulness of those definitions is the ease with which we can solve many problems using conservation of energy. Potential energy is particularly useful for forces that change with position, as the gravitational force does over large distances. In Potential Energy and Conservation of Energy, we showed that the change in gravitational potential energy near Earth’s surface is $\text{Δ}U=mg\left({y}_{2}-{y}_{1}\right)$. This works very well if *g* does not change significantly between ${y}_{1}$ and ${y}_{2}$. We return to the definition of work and potential energy to derive an expression that is correct over larger distances.

Conservation of Energy

In Potential Energy and Conservation of Energy, we described how to apply conservation of energy for systems with conservative forces. We were able to solve many problems, particularly those involving gravity, more simply using conservation of energy. Those principles and problem-solving strategies apply equally well here. The only change is to place the new expression for potential energy into the conservation of energy equation, $E={K}_{1}+{U}_{1}={K}_{2}+{U}_{2}$.

Satellite Orbits and Energy

Abstract:
By the end of this section, you will be able to:

* Describe the mechanism for circular orbits
* Find the orbital periods and speeds of satellites
* Determine whether objects are gravitationally bound

Circular Orbits

As noted at the beginning of this chapter, Nicolaus Copernicus first suggested that Earth and all other planets orbit the Sun in circles. He further noted that orbital periods increased with distance from the Sun. Later analysis by Kepler showed that these orbits are actually ellipses, but the orbits of most planets in the solar system are nearly circular. Earth’s orbital distance from the Sun varies a mere 2%. The exception is the eccentric orbit of Mercury, whose orbital distance varies nearly 40%.

Energy in Circular Orbits

In Gravitational Potential Energy and Total Energy, we argued that objects are gravitationally bound if their total energy is negative. The argument was based on the simple case where the velocity was directly away or toward the planet. We now examine the total energy for a circular orbit and show that indeed, the total energy is negative. As we did earlier, we start with Newton’s second law applied to a circular orbit,

Kepler's Laws of Planetary Motion

Abstract:
By the end of this section, you will be able to:

* Describe the conic sections and how they relate to orbital motion
* Describe how orbital velocity is related to conservation of angular momentum
* Determine the period of an elliptical orbit from its major axis

Kepler’s First Law

The prevailing view during the time of Kepler was that all planetary orbits were circular. The data for Mars presented the greatest challenge to this view and that eventually encouraged Kepler to give up the popular idea. Kepler’s first law states that every planet moves along an ellipse, with the Sun located at a focus of the ellipse. An ellipse is defined as the set of all points such that the sum of the distance from each point to two foci is a constant. [link] shows an ellipse and describes a simple way to create it.

Orbital Transfers

People have imagined traveling to the other planets of our solar system since they were discovered. But how can we best do this? The most efficient method was discovered in 1925 by Walter Hohmann, inspired by a popular science fiction novel of that time. The method is now called a Hohmann transfer. For the case of traveling between two circular orbits, the transfer is along a “transfer” ellipse that perfectly intercepts those orbits at the aphelion and perihelion of the ellipse. [link] shows the case for a trip from Earth’s orbit to that of Mars. As before, the Sun is at the focus of the ellipse.

Kepler’s Second Law

Kepler’s second law states that a planet sweeps out equal areas in equal times, that is, the area divided by time, called the areal velocity, is constant. Consider [link]. The time it takes a planet to move from position *A* to *B*, sweeping out area ${A}_{1}$, is exactly the time taken to move from position *C* to *D*, sweeping area ${A}_{2}$, and to move from *E* to *F*, sweeping out area ${A}_{3}$. These areas are the same: ${A}_{1}={A}_{2}={A}_{3}$.

Kepler’s Third Law

Kepler’s third law states that the square of the period is proportional to the cube of the semi-major axis of the orbit. In Satellite Orbits and Energy, we derived Kepler’s third law for the special case of a circular orbit. [link] gives us the period of a circular orbit of radius *r* about Earth:

Tidal Forces

Abstract:
By the end of this section, you will be able to:

* Explain the origins of Earth’s ocean tides
* Describe how neap and leap tides differ
* Describe how tidal forces affect binary systems

Lunar Tides

If you live on an ocean shore almost anywhere in the world, you can observe the rising and falling of the sea level about twice per day. This is caused by a combination of Earth’s rotation about its axis and the gravitational attraction of both the Moon and the Sun.

The Effect of the Sun on Tides

In addition to the Moon’s tidal forces on Earth’s oceans, the Sun exerts a tidal force as well. The gravitational attraction of the Sun on any object on Earth is nearly 200 times that of the Moon. However, as we show later in an example, the *tidal* effect of the Sun is less than that of the Moon, but a significant effect nevertheless. Depending upon the positions of the Moon and Sun relative to Earth, the net tidal effect can be amplified or attenuated.

The Magnitude of the Tides

With accurate data for the positions of the Moon and the Sun, the time of maximum and minimum tides at most locations on our planet can be predicted accurately.

Other Tidal Effects

Tidal forces exist between any two bodies. The effect stretches the bodies along the line between their centers. Although the tidal effect on Earth’s seas is observable on a daily basis, long-term consequences cannot be observed so easily. One consequence is the dissipation of rotational energy due to friction during flexure of the bodies themselves. Earth’s rotation rate is slowing down as the tidal forces transfer rotational energy into heat. The other effect, related to this dissipation and conservation of angular momentum, is called “locking” or tidal synchronization. It has already happened to most moons in our solar system, including Earth’s Moon. The Moon keeps one face toward Earth—its rotation rate has locked into the orbital rate about Earth. The same process is happening to Earth, and eventually it will keep one face toward the Moon. If that does happen, we would no longer see tides, as the tidal bulge would remain in the same place on Earth, and half the planet would never see the Moon. However, this locking will take many billions of years, perhaps not before our Sun expires.

Einstein's Theory of Gravity

Abstract:
By the end of this section, you will be able to:

* Describe how the theory of general relativity approaches gravitation
* Explain the principle of equivalence
* Calculate the Schwarzschild radius of an object
* Summarize the evidence for black holes

A Revolution in Perspective

In 1905, Albert Einstein published his theory of special relativity. This theory is discussed in great detail in Relativity, so we say only a few words here. In this theory, no motion can exceed the speed of light—it is the speed limit of the Universe. This simple fact has been verified in countless experiments. However, it has incredible consequences—space and time are no longer absolute. Two people moving relative to one another do not agree on the length of objects or the passage of time. Almost all of the mechanics you learned in previous chapters, while remarkably accurate even for speeds of many thousands of miles per second, begin to fail when approaching the speed of light.

The Principle of Equivalence

Einstein came to his general theory in part by wondering why someone who was free falling did not feel their weight. Indeed, it is common to speak of astronauts orbiting Earth as being weightless, despite the fact that Earth’s gravity is still quite strong there. In Einstein’s general theory, there is no difference between free fall and being weightless. This is called the principle of equivalence. The equally surprising corollary to this is that there is no difference between a uniform gravitational field and a uniform acceleration in the absence of gravity. Let’s focus on this last statement. Although a perfectly uniform gravitational field is not feasible, we can approximate it very well.

A Geometric Theory of Gravity

Euclidian geometry assumes a “flat” space in which, among the most commonly known attributes, a straight line is the shortest distance between two points, the sum of the angles of all triangles must be 180 degrees, and parallel lines never intersect. Non-Euclidean geometry was not seriously investigated until the nineteenth century, so it is not surprising that Euclidean space is inherently assumed in all of Newton’s laws.

Black Holes

Einstein’s theory of gravitation is expressed in one deceptively simple-looking tensor equation (tensors are a generalization of scalars and vectors), which expresses how a mass determines the curvature of space-time around it. The solutions to that equation yield one of the most fascinating predictions: the black hole. The prediction is that if an object is sufficiently dense, it will collapse in upon itself and be surrounded by an event horizon from which nothing can escape. The name “black hole,” which was coined by astronomer John Wheeler in 1969, refers to the fact that light cannot escape such an object. Karl Schwarzschild was the first person to note this phenomenon in 1916, but at that time, it was considered mostly to be a mathematical curiosity.

Fluid Mechanics

Introduction

Fluids, Density, and Pressure

Abstract:
By the end of this section, you will be able to:

* State the different phases of matter
* Describe the characteristics of the phases of matter at the molecular or atomic level
* Distinguish between compressible and incompressible materials
* Define density and its related SI units
* Compare and contrast the densities of various substances
* Define pressure and its related SI units
* Explain the relationship between pressure and force
* Calculate force given pressure and area

Characteristics of Solids

Solids are rigid and have specific shapes and definite volumes. The atoms or molecules in a solid are in close proximity to each other, and there is a significant force between these molecules. Solids will take a form determined by the nature of these forces between the molecules. Although true solids are not incompressible, it nevertheless requires a large force to change the shape of a solid. In some cases, the force between molecules can cause the molecules to organize into a lattice as shown in [link]. The structure of this three-dimensional lattice is represented as molecules connected by rigid bonds (modeled as stiff springs), which allow limited freedom for movement. Even a large force produces only small displacements in the atoms or molecules of the lattice, and the solid maintains its shape. Solids also resist shearing forces. (Shearing forces are forces applied tangentially to a surface, as described in Static Equilibrium and Elasticity.)

Characteristics of Fluids

Liquids and gases are
 considered to be fluids because they yield to shearing forces, whereas solids resist them. Like solids, the molecules in a liquid are bonded to neighboring molecules, but possess many fewer of these bonds. The molecules in a liquid are not locked in place and can move with respect to each other. The distance between molecules is similar to the distances in a solid, and so liquids have definite volumes, but the shape of a liquid changes, depending on the shape of its container. Gases are not bonded to neighboring atoms and can have large separations between molecules. Gases have neither specific shapes nor definite volumes, since their molecules move to fill the container in which they are held ([link]).

Density

Suppose a block of brass and a block of wood have exactly the same mass. If both blocks are dropped in a tank of water, why does the wood float and the brass sink ([link])? This occurs because the brass has a greater density than water, whereas the wood has a lower density than water.

Pressure

You have no doubt heard the word ‘pressure’ used in relation to blood (high or low blood pressure) and in relation to weather (high- and low-pressure weather systems). These are only two of many examples of pressure in fluids. (Recall that we introduced the idea of pressure in Static Equilibrium and Elasticity, in the context of bulk stress and strain.)

Measuring Pressure

Abstract:
By the end of this section, you will be able to:

* Define gauge pressure and absolute pressure
* Explain various methods for measuring pressure
* Understand the working of open-tube barometers
* Describe in detail how manometers and barometers operate

Gauge Pressure vs. Absolute Pressure

Suppose the pressure gauge on a full scuba tank reads 3000 psi, which is approximately 207 atmospheres. When the valve is opened, air begins to escape because the pressure inside the tank is greater than the atmospheric pressure outside the tank. Air continues to escape from the tank until the pressure inside the tank equals the pressure of the atmosphere outside the tank. At this point, the pressure gauge on the tank reads zero, even though the pressure inside the tank is actually 1 atmosphere—the same as the air pressure outside the tank.

Measuring Pressure

A host of devices are used for measuring pressure, ranging from tire gauges to blood pressure monitors. Many other types of pressure gauges are commonly used to test the pressure of fluids, such as mechanical pressure gauges. We will explore some of these in this section.

Pascal's Principle and Hydraulics

Abstract:
By the end of this section, you will be able to:

* State Pascal’s principle
* Describe applications of Pascal’s principle
* Derive relationships between forces in a hydraulic system

Pascal’s Principle

Pascal’s principle (also known as Pascal’s law) states that when a change in pressure is applied to an enclosed fluid, it is transmitted undiminished to all portions of the fluid and to the walls of its container. In an enclosed fluid, since atoms of the fluid are free to move about, they transmit pressure to all parts of the fluid *and* to the walls of the container. Any change in pressure is transmitted undiminished.

Applications of Pascal’s Principle and Hydraulic Systems

Hydraulic systems are used to operate automotive brakes, hydraulic jacks, and numerous other mechanical systems ([link]).

Archimedes’ Principle and Buoyancy

Abstract:
By the end of this section, you will be able to:

* Define buoyant force
* State Archimedes’ principle
* Describe the relationship between density and Archimedes’ principle

Archimedes’ Principle

Just how large a force is buoyant force? To answer this question, think about what happens when a submerged object is removed from a fluid, as in [link]. If the object were not in the fluid, the space the object occupied would be filled by fluid having a weight ${w}_{\text{fl}}.$ This weight is supported by the surrounding fluid, so the buoyant force must equal ${w}_{\text{fl}},$ the weight of the fluid displaced by the object.

Density and Archimedes’ Principle

If you drop a lump of clay in water, it will sink. But if you mold the same lump of clay into the shape of a boat, it will float. Because of its shape, the clay boat displaces more water than the lump and experiences a greater buoyant force, even though its mass is the same. The same is true of steel ships.

Measuring Density

One of the most common techniques for determining density is shown in [link].

Fluid Dynamics

Abstract:
By the end of this section, you will be able to:

* Describe the characteristics of flow
* Calculate flow rate
* Describe the relationship between flow rate and velocity
* Explain the consequences of the equation of continuity to the conservation of mass

Characteristics of Flow

Velocity vectors are often used to illustrate fluid motion in applications like meteorology. For example, wind—the fluid motion of air in the atmosphere—can be represented by vectors indicating the speed and direction of the wind at any given point on a map. [link] shows velocity vectors describing the winds during Hurricane Arthur in 2014.

Flow Rate and its Relation to Velocity

The volume of fluid passing by a given location through an area during a period of time is called flow rate *Q*, or more precisely, volume flow rate. In symbols, this is written as

Mass Conservation

The rate of flow of a fluid can also be described by the *mass flow rate* or mass rate of flow. This is the rate at which a mass of the fluid moves past a point. Refer once again to [link], but this time consider the mass in the shaded volume. The mass can be determined from the density and the volume:

Bernoulli’s Equation

Abstract:
By the end of this section, you will be able to:

* Explain the terms in Bernoulli’s equation
* Explain how Bernoulli’s equation is related to the conservation of energy
* Describe how to derive Bernoulli’s principle from Bernoulli’s equation
* Perform calculations using Bernoulli’s principle
* Describe some applications of Bernoulli’s principle

Energy Conservation and Bernoulli’s Equation

The application of the principle of conservation of energy to frictionless laminar flow leads to a very useful relation between pressure and flow speed in a fluid. This relation is called Bernoulli’s equation, named after Daniel Bernoulli (1700–1782), who published his studies on fluid motion in his book *Hydrodynamica* (1738).

Analyzing Bernoulli’s Equation

According to Bernoulli’s equation, if we follow a small volume of fluid along its path, various quantities in the sum may change, but the total remains constant. Bernoulli’s equation is, in fact, just a convenient statement of conservation of energy for an incompressible fluid in the absence of friction.

Applications of Bernoulli’s Principle

Many devices and situations occur in which fluid flows at a constant height and thus can be analyzed with Bernoulli’s principle.

Viscosity and Turbulence

Abstract:
By the end of this section, you will be able to:

* Explain what viscosity is
* Calculate flow and resistance with Poiseuille's law
* Explain how pressure drops due to resistance
* Calculate the Reynolds number for an object moving through a fluid
* Use the Reynolds number for a system to determine whether it is laminar or turbulent
* Describe the conditions under which an object has a terminal speed

Viscosity and Laminar Flow

When you pour yourself a glass of juice, the liquid flows freely and quickly. But if you pour maple syrup on your pancakes, that liquid flows slowly and sticks to the pitcher. The difference is fluid friction, both within the fluid itself and between the fluid and its surroundings. We call this property of fluids *viscosity*. Juice has low viscosity, whereas syrup has high viscosity.

Laminar Flow Confined to Tubes: Poiseuille’s Law

What causes flow? The answer, not surprisingly, is a pressure difference. In fact, there is a very simple relationship between horizontal flow and pressure. Flow rate *Q* is in the direction from high to low pressure. The greater the pressure differential between two points, the greater the flow rate. This relationship can be stated as

Flow and Resistance as Causes of Pressure Drops

Water pressure in homes is sometimes lower than normal during times of heavy use, such as hot summer days. The drop in pressure occurs in the water main before it reaches individual homes. Let us consider flow through the water main as illustrated in [link]. We can understand why the pressure ${p}_{1}$ to the home drops during times of heavy use by rearranging the equation for flow rate:

Measuring Turbulence

An indicator called the Reynolds number ${N}_{\text{R}}$ can reveal whether flow is laminar or turbulent. For flow in a tube of uniform diameter, the Reynolds number is defined as

Waves and Acoustics

Oscillations

Introduction

Simple Harmonic Motion

Abstract:
By the end of this section, you will be able to:

* Define the terms period and frequency
* List the characteristics of simple harmonic motion
* Explain the concept of phase shift
* Write the equations of motion for the system of a mass and spring undergoing simple harmonic motion
* Describe the motion of a mass oscillating on a vertical spring

Period and Frequency in Oscillations

In the absence of friction, the time to complete one oscillation remains constant and is called the period (*T*). Its units are usually seconds, but may be any convenient unit of time. The word ‘period’ refers to the time for some event whether repetitive or not, but in this chapter, we shall deal primarily in periodic motion, which is by definition repetitive.

Characteristics of Simple Harmonic Motion

A very common type of periodic motion is called simple harmonic motion (SHM). A system that oscillates with SHM is called a simple harmonic oscillator.

Equations of SHM

Consider a block attached to a spring on a frictionless table ([link]). The equilibrium position (the position where the spring is neither stretched nor compressed) is marked as $x=0$. At the equilibrium position, the net force is zero.

Summary of Equations of Motion for SHM

In summary, the oscillatory motion of a block on a spring can be modeled with the following equations of motion:

The Period and Frequency of a Mass on a Spring

One interesting characteristic of the SHM of an object attached to a spring is that the angular frequency, and therefore the period and frequency of the motion, depend on only the mass and the force constant, and not on other factors such as the amplitude of the motion. We can use the equations of motion and Newton’s second law $\left({\stackrel{\to }{F}}_{\text{net}}=m\stackrel{\to }{a}\right)$ to find equations for the angular frequency, frequency, and period.

Vertical Motion and a Horizontal Spring

When a spring is hung vertically and a block is attached and set in motion, the block oscillates in SHM. In this case, there is no normal force, and the net effect of the force of gravity is to change the equilibrium position. Consider [link]. Two forces act on the block: the weight and the force of the spring. The weight is constant and the force of the spring changes as the length of the spring changes.

Energy in Simple Harmonic Motion

Abstract:
By the end of this section, you will be able to:

* Describe the energy conservation of the system of a mass and a spring
* Explain the concepts of stable and unstable equilibrium points

Energy and the Simple Harmonic Oscillator

To study the energy of a simple harmonic oscillator, we need to consider all the forms of energy. Consider the example of a block attached to a spring, placed on a frictionless surface, oscillating in SHM. The potential energy stored in the deformation of the spring is

Oscillations About an Equilibrium Position

We have just considered the energy of SHM as a function of time. Another interesting view of the simple harmonic oscillator is to consider the energy as a function of position. [link] shows a graph of the energy versus position of a system undergoing SHM.

Velocity and Energy Conservation

Getting back to the system of a block and a spring in [link], once the block is released from rest, it begins to move in the negative direction toward the equilibrium position. The potential energy decreases and the magnitude of the velocity and the kinetic energy increase. At time $t=T\text{/}4$, the block reaches the equilibrium position $x=0.00{\rule{0.2em}{0ex}}\text{m,}$ where the force on the block and the potential energy are zero. At the equilibrium position, the block reaches a negative velocity with a magnitude equal to the maximum velocity $v=\text{−}A\omega$. The kinetic energy is maximum and equal to $K=\frac{1}{2}m{v}^{2}=\frac{1}{2}m{A}^{2}{\omega }^{2}=\frac{1}{2}k{A}^{2}.$ At this point, the force on the block is zero, but momentum carries the block, and it continues in the negative direction toward $x=\text{−}A$. As the block continues to move, the force on it acts in the positive direction and the magnitude of the velocity and kinetic energy decrease. The potential energy increases as the spring compresses. At time $t=T\text{/}2$, the block reaches $x=\text{−}A$. Here the velocity and kinetic energy are equal to zero. The force on the block is $F=+kA$ and the potential energy stored in the spring is $U=\frac{1}{2}k{A}^{2}$. During the oscillations, the total energy is constant and equal to the sum of the potential energy and the kinetic energy of the system,

Comparing Simple Harmonic Motion and Circular Motion

Abstract:
By the end of this section, you will be able to:

* Describe how the sine and cosine functions relate to the concepts of circular motion
* Describe the connection between simple harmonic motion and circular motion

Pendulums

Abstract:
By the end of this section, you will be able to:

* State the forces that act on a simple pendulum
* Determine the angular frequency, frequency, and period of a simple pendulum in terms of the length of the pendulum and the acceleration due to gravity
* Define the period for a physical pendulum
* Define the period for a torsional pendulum

The Simple Pendulum

A simple pendulum is defined to have a point mass, also known as the pendulum bob, which is suspended from a string of length *L* with negligible mass ([link]). Here, the only forces acting on the bob are the force of gravity (i.e., the weight of the bob) and tension from the string. The mass of the string is assumed to be negligible as compared to the mass of the bob.

Physical Pendulum

Any object can oscillate like a pendulum. Consider a coffee mug hanging on a hook in the pantry. If the mug gets knocked, it oscillates back and forth like a pendulum until the oscillations die out. We have described a simple pendulum as a point mass and a string. A physical pendulum is any object whose oscillations are similar to those of the simple pendulum, but cannot be modeled as a point mass on a string, and the mass distribution must be included into the equation of motion.

Torsional Pendulum

A torsional pendulum consists of a rigid body suspended by a light wire or spring ([link]). When the body is twisted some small maximum angle $\left(\text{Θ}\right)$ and released from rest, the body oscillates between $\left(\theta =+\text{Θ}\right)$ and $\left(\theta =\text{−}{\rule{0.2em}{0ex}}\text{Θ}\right)$. The restoring torque is supplied by the shearing of the string or wire.

Damped Oscillations

Abstract:
By the end of this section, you will be able to:

* Describe the motion of damped harmonic motion
* Write the equations of motion for damped harmonic oscillations
* Describe the motion of driven, or forced, damped harmonic motion
* Write the equations of motion for forced, damped harmonic motion

Forced Oscillations

Abstract:
By the end of this section, you will be able to:

* Define forced oscillations
* List the equations of motion associated with forced oscillations
* Explain the concept of resonance and its impact on the amplitude of an oscillator
* List the characteristics of a system oscillating in resonance

Waves

Introduction

Traveling Waves

Abstract:
By the end of this section, you will be able to:

* Describe the basic characteristics of wave motion
* Define the terms wavelength, amplitude, period, frequency, and wave speed
* Explain the difference between longitudinal and transverse waves, and give examples of each type
* List the different types of waves

Types of Waves

A wave is a disturbance that propagates, or moves from the place it was created. There are three basic types of waves: mechanical waves, electromagnetic waves, and matter waves.

Mechanical Waves

Mechanical waves exhibit characteristics common to all waves, such as amplitude, wavelength, period, frequency, and energy. All wave characteristics can be described by a small set of underlying principles.

Transverse and Longitudinal Waves

We have seen that a simple mechanical wave consists of a periodic disturbance that propagates from one place to another through a medium. In [link](a), the wave propagates in the horizontal direction, whereas the medium is disturbed in the vertical direction. Such a wave is called a transverse wave. In a transverse wave, the wave may propagate in any direction, but the disturbance of the medium is perpendicular to the direction of propagation. In contrast, in a longitudinal wave or compressional wave, the disturbance is parallel to the direction of propagation. [link](b) shows an example of a longitudinal wave. The size of the disturbance is its amplitude *A* and is completely independent of the speed of propagation *v*.

Mathematics of Waves

Abstract:
By the end of this section, you will be able to:

* Model a wave, moving with a constant wave velocity, with a mathematical expression
* Calculate the velocity and acceleration of the medium
* Show how the velocity of the medium differs from the wave velocity (propagation velocity)

Pulses

A pulse can be described as wave consisting of a single disturbance that moves through the medium with a constant amplitude. The pulse moves as a pattern that maintains its shape as it propagates with a constant wave speed. Because the wave speed is constant, the distance the pulse moves in a time $\text{Δ}t$ is equal to $\text{Δ}x=v\text{Δ}t$ ([link]).

Modeling a One-Dimensional Sinusoidal Wave using a Wave Function

Consider a string kept at a constant tension ${F}_{T}$ where one end is fixed and the free end is oscillated between $y=\text{+}A$ and $y=\text{−}A$ by a mechanical device at a constant frequency. [link] shows snapshots of the wave at an interval of an eighth of a period, beginning after one period $\left(t=T\right).$

Velocity and Acceleration of the Medium

As seen in [link], the wave speed is constant and represents the speed of the wave as it propagates through the medium, not the speed of the particles that make up the medium. The particles of the medium oscillate around an equilibrium position as the wave propagates through the medium. In the case of the transverse wave propagating in the *x*-direction, the particles oscillate up and down in the *y*-direction, perpendicular to the motion of the wave. The velocity of the particles of the medium is not constant, which means there is an acceleration. The velocity of the medium, which is perpendicular to the wave velocity in a transverse wave, can be found by taking the partial derivative of the position equation with respect to time. The partial derivative is found by taking the derivative of the function, treating all variables as constants, except for the variable in question. In the case of the partial derivative with respect to time *t*, the position *x* is treated as a constant. Although this may sound strange if you haven’t seen it before, the object of this exercise is to find the transverse velocity at a point, so in this sense, the *x*-position is not changing. We have

The Linear Wave Equation

We have just determined the velocity of the medium at a position *x* by taking the partial derivative, with respect to time, of the position *y*. For a transverse wave, this velocity is perpendicular to the direction of propagation of the wave. We found the acceleration by taking the partial derivative, with respect to time, of the velocity, which is the second time derivative of the position:

Wave Speed on a Stretched String

Abstract:
By the end of this section, you will be able to:

* Determine the factors that affect the speed of a wave on a string
* Write a mathematical expression for the speed of a wave on a string and generalize these concepts for other media

Wave Speed on a String under Tension

To see how the speed of a wave on a string depends on the tension and the linear density, consider a pulse sent down a taut string ([link]). When the taut string is at rest at the equilibrium position, the tension in the string ${F}_{T}$ is constant. Consider a small element of the string with a mass equal to $\text{Δ}m=\mu \text{Δ}x.$ The mass element is at rest and in equilibrium and the force of tension of either side of the mass element is equal and opposite.

Speed of Compression Waves in a Fluid

The speed of a wave on a string depends on the square root of the tension divided by the mass per length, the linear density. In general, the speed of a wave through a medium depends on the elastic property of the medium and the inertial property of the medium.

Energy and Power of a Wave

Abstract:
By the end of this section, you will be able to:

* Explain how energy travels with a pulse or wave
* Describe, using a mathematical expression, how the energy in a wave depends on the amplitude of the wave

Energy in Waves

The amount of energy in a wave is related to its amplitude and its frequency. Large-amplitude earthquakes produce large ground displacements. Loud sounds have high-pressure amplitudes and come from larger-amplitude source vibrations than soft sounds. Large ocean breakers churn up the shore more than small ones. Consider the example of the seagull and the water wave earlier in the chapter ([link]). Work is done on the seagull by the wave as the seagull is moved up, changing its potential energy. The larger the amplitude, the higher the seagull is lifted by the wave and the larger the change in potential energy.

Power in Waves

Consider a sinusoidal wave on a string that is produced by a string vibrator, as shown in [link]. The string vibrator is a device that vibrates a rod up and down. A string of uniform linear mass density is attached to the rod, and the rod oscillates the string, producing a sinusoidal wave. The rod does work on the string, producing energy that propagates along the string. Consider a mass element of the string with a mass $\text{Δ}m$, as seen in [link]. As the energy propagates along the string, each mass element of the string is driven up and down at the same frequency as the wave. Each mass element of the string can be modeled as a simple harmonic oscillator. Since the string has a constant linear density $\mu =\frac{\text{Δ}m}{\text{Δ}x},$ each mass element of the string has the mass $\text{Δ}m=\mu \text{Δ}x.$

Interference of Waves

Abstract:
By the end of this section, you will be able to:

* Explain how mechanical waves are reflected and transmitted at the boundaries of a medium
* Define the terms interference and superposition
* Find the resultant wave of two identical sinusoidal waves that differ only by a phase shift

Reflection and Transmission

When a wave propagates through a medium, it reflects when it encounters the boundary of the medium. The wave before hitting the boundary is known as the incident wave. The wave after encountering the boundary is known as the reflected wave. How the wave is reflected at the boundary of the medium depends on the boundary conditions; waves will react differently if the boundary of the medium is fixed in place or free to move ([link]). A fixed boundary condition exists when the medium at a boundary is fixed in place so it cannot move. A free boundary condition exists when the medium at the boundary is free to move.

Superposition and Interference

Most waves do not look very simple. Complex waves are more interesting, even beautiful, but they look formidable. Most interesting mechanical waves consist of a combination of two or more traveling waves propagating in the same medium. The principle of superposition can be used to analyze the combination of waves.

Superposition of Sinusoidal Waves that Differ by a Phase Shift

Many examples in physics consist of two sinusoidal waves that are identical in amplitude, wave number, and angular frequency, but differ by a phase shift:

Standing Waves and Resonance

Abstract:
By the end of this section, you will be able to:

* Describe standing waves and explain how they are produced
* Describe the modes of a standing wave on a string
* Provide examples of standing waves beyond the waves on a string

Standing Waves

Sometimes waves do not seem to move; rather, they just vibrate in place. You can see unmoving waves on the surface of a glass of milk in a refrigerator, for example. Vibrations from the refrigerator motor create waves on the milk that oscillate up and down but do not seem to move across the surface. [link] shows an experiment you can try at home. Take a bowl of milk and place it on a common box fan. Vibrations from the fan will produce circular standing waves in the milk. The waves are visible in the photo due to the reflection from a lamp. These waves are formed by the superposition of two or more traveling waves, such as illustrated in [link] for two identical 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. The resultant looks like a wave standing in place and, thus, is called a standing wave.

Sound

Introduction

Sound Waves

Abstract:
By the end of this section, you will be able to:

* Explain the difference between sound and hearing
* Describe sound as a wave
* List the equations used to model sound waves
* Describe compression and rarefactions as they relate to sound

Speed of Sound

Abstract:
By the end of this section, you will be able to:

* Explain the relationship between wavelength and frequency of sound
* Determine the speed of sound in different media
* Derive the equation for the speed of sound in air
* Determine the speed of sound in air for a given temperature

Speed of Sound in Various Media

[link] shows that the speed of sound varies greatly in different media. The speed of sound in a medium depends on how quickly vibrational energy can be transferred through the medium. For this reason, the derivation of the speed of sound in a medium depends on the medium and on the state of the medium. In general, the equation for the speed of a mechanical wave in a medium depends on the square root of the restoring force, or the elastic property, divided by the inertial property,

Derivation of the Speed of Sound in Air

As stated earlier, the speed of sound in a medium depends on the medium and the state of the medium. The derivation of the equation for the speed of sound in air starts with the mass flow rate and continuity equation discussed in Fluid Mechanics.

Sound Intensity

Abstract:
By the end of this section, you will be able to:

* Define the term intensity
* Explain the concept of sound intensity level
* Describe how the human ear translates sound

Human Hearing and Sound Intensity Levels

As stated earlier in this chapter, hearing is the perception of sound. The hearing mechanism involves some interesting physics. The sound wave that impinges upon our ear is a pressure wave. The ear is a transducer that converts sound waves into electrical nerve impulses in a manner much more sophisticated than, but analogous to, a microphone. [link] shows the anatomy of the ear.

Hearing and Pitch

The human ear has a tremendous range and sensitivity. It can give us a wealth of simple information—such as pitch, loudness, and direction.

Normal Modes of a Standing Sound Wave

Abstract:
By the end of this section, you will be able to:

* Explain the mechanism behind sound-reducing headphones
* Describe resonance in a tube closed at one end and open at the other end
* Describe resonance in a tube open at both ends

Interference of Sound Waves

In Waves, we discussed the interference of wave functions that differ only in a phase shift. We found that the wave function resulting from the superposition of ${y}_{1}\left(x,t\right)=A{\rule{0.2em}{0ex}}\text{sin}\left(kx-\omega t+\varphi \right)$ and ${y}_{2}\left(x,t\right)=A{\rule{0.2em}{0ex}}\text{sin}\left(kx-\omega t\right)$ is

Noise Reduction through Destructive Interference

[link] shows a clever use of sound interference to cancel noise. Larger-scale applications of active noise reduction by destructive interference have been proposed for entire passenger compartments in commercial aircraft. To obtain destructive interference, a fast electronic analysis is performed, and a second sound is introduced $180\text{°}$ out of phase with the original sound, with its maxima and minima exactly reversed from the incoming noise. Sound waves in fluids are pressure waves and are consistent with Pascal’s principle; that is, pressures from two different sources add and subtract like simple numbers. Therefore, positive and negative gauge pressures add to a much smaller pressure, producing a lower-intensity sound. Although completely destructive interference is possible only under the simplest conditions, it is possible to reduce noise levels by 30 dB or more using this technique.

Resonance in a Tube Closed at one End

As we discussed in Waves, *standing waves* are formed by two waves moving in opposite directions. When two identical sinusoidal waves move in opposite directions, the waves may be modeled as

Resonance in a Tube Open at Both Ends

Another source of standing waves is a tube that is open at both ends. In this case, the boundary conditions are symmetrical: an antinode at each end. The resonances of tubes open at both ends can be analyzed in a very similar fashion to those for tubes closed at one end. The air columns in tubes open at both ends have maximum air displacements at both ends ([link]). Standing waves form as shown.

Sources of Musical Sound

Abstract:
By the end of this section, you will be able to:

* Describe the resonant frequencies in instruments that can be modeled as a tube with symmetrical boundary conditions
* Describe the resonant frequencies in instruments that can be modeled as a tube with anti-symmetrical boundary conditions

Resonance

Resonance occurs in many different systems, including strings, air columns, and atoms. As we discussed in earlier chapters, resonance is the driven or forced oscillation of a system at its natural frequency. At resonance, energy is transferred rapidly to the oscillating system, and the amplitude of its oscillations grows until the system can no longer be described by Hooke’s law. An example of this is the distorted sound intentionally produced in certain types of rock music.

Beats

Abstract:
By the end of this section, you will be able to:

* Determine the beat frequency produced by two sound waves that differ in frequency
* Describe how beats are produced by musical instruments

The Doppler Effect

Abstract:
By the end of this section, you will be able to:

* Explain the change in observed frequency as a moving source of sound approaches or departs from a stationary observer
* Explain the change in observed frequency as an observer moves toward or away from a stationary source of sound

Derivation of the Observed Frequency due to the Doppler Shift

Consider two stationary observers *X* and *Y* in [link], located on either side of a stationary source. Each observer hears the same frequency, and that frequency is the frequency produced by the stationary source.

Shock Waves

Abstract:
By the end of this section, you will be able to:

* Explain the mechanism behind sonic booms
* Describe the difference between sonic booms and shock waves
* Describe a bow wake

Doppler Effect and High Velocity

What happens to the sound produced by a moving source, such as a jet airplane, that approaches or even exceeds the speed of sound? The answer to this question applies not only to sound but to all other waves as well. Suppose a jet plane is coming nearly straight at you, emitting a sound of frequency ${f}_{\text{s}}.$ The greater the plane’s speed ${v}_{\text{s}},$ the greater the Doppler shift and the greater the value observed for ${f}_{\text{o}}$ ([link]).

Shock Waves and Sonic Booms

If the source exceeds the speed of sound, no sound is received by the observer until the source has passed, so that the sounds from the approaching source are mixed with those from it when receding. This mixing appears messy, but something interesting happens—a shock wave is created ([link]).

Units

Conversion Factors

Fundamental Constants

Astronomical Data

Mathematical Formulas

Chemistry

The Greek Alphabet