# Spinning Stopper Lab

Research Question: How does the time period of the stopper depend upon the length of the string? Independent variable: Length of String (m) 0.05m, 0.10m, 0.15m, 0.20m and 0.25m Dependant Variable: Time Period (s) Control Variables: Mass of stopper – This must be constant as we are measuring the relationship between the time period and length of string. The mass of the stopper should not change as, for example, a larger mass would mean a larger tension force on the string, changing the time period for that particular string length. I will used the same bung throughout the experiment, with a mass of 12.70g. Mass hanging on string – I will used the same mass (27.5g) throughout all of the trials because it is a constant in the experiment, and would change the time period. Method of spinning – the method of spinning should be such that the circle remains horizontal. This will ensure that the laws of uniform circular motion can be applied. Friction in system – It is difficult to control the friction in the system, however if the same length of string and plastic tube is used (the objects in the experiment that will cause friction), then the friction should also remain constant. Properties of string – The same string should be used for all of the trials, as the properties of different types of string changes; some may be more elastic or generate more friction than others.

The method below will control these variables, so that they do not affect the data collected from the dependant variable. Apparatus   Rubber stopper/bung String     Plastic tube (casing of a pen) Mass (higher mass than bung) 30cm Ruler Stopwatch

Diagram

1

Diagram 1 Source:www.physics-inthinking.co.uk

Method 1. Tie the bung to one end of the string 2. Using the ruler mark 5cm intervals along the string starting at the end tied to the bung. Do this until you have made a total of 5 marks, reaching 25cm along the string. 3. Set up the experiment as shown in Diagram 1, with the rubber stopper tied to the end of the string, with the other end threaded through the tube with the mass attached to the bottom. 4. Keeping the tube vertical, practise swinging the bung around with as little movement as possible from your hand, and without holding the string with mass attached at the bottom. You should do this until you can keep the frequency constant and the circle horizontal. 5. Simultaneously start the stopwatch whilst spinning the rubber bung, and time how long it takes for 20 rotations at a string length of 0.05m. Record the time in seconds in the data table. 6. Repeat for 5 trials, recording the time in second for each. 7. Repeat points 5 and 6 for all of the independent variable values (String lengths: 0.05m, 0.10m, 0.15m, 0.20m and 0.25m).

Raw Data Collection Mass of stopper: 12.70g (0.0127kg) (M) Mass of weight: 27.5g (0.0275kg) (m)
Table 1

Length of String ± Rotations 0.02 (m) 0.05 20 0.10 20 0.15 20 0.20 20 0.25 20

Time 1 ± 0.1 (s) 6.8 9.1 10.1 11.5 12.5

Time 2 ± 0.1 (s) 6.8 8.9 10.6 12.1 13.2

Time 3 ± 0.1 (s) 6.6 8.9 10.2 11.5 12.9

Time 4 ± 0.1 (s) 6.7 8.7 10.1 11.4 13.1

Time 5 ± 0.1 (s) 6.8 9.0 10.3 11.7 12.8

Table 1 below shows the raw data that was collected during the experiment. All data is recorded in SI units of measurement: second and metres. I have estimated the uncertainties in time due to systematic error, based on the limit of accuracy of the stopwatches: they were only precise to the nearest 0.01 seconds. The systematic error I have estimated also accounts for the human reaction 2

time of stopping the stopwatch at the right moment, giving the value of ± 0.1 (s). The uncertainty in the length of string was estimated at ± 0.02 (m), due to the fact that when the bung was rotating, because the length of string was dependent upon the velocity, it varied slightly, which I concluded to be at a maximum of ± 0.02 (m).
Graph 1

Length of String (m) vs Time (s)
14 13 12 11

Time (s)

10 9 8 7 6 5 0 0.05 0.1 0.15 0.2 0.25 0.3

Length of String (m)
Time 1 ± 0.1 (s) Time 2 ± 0.1 (s) Time 3 ± 0.1 (s) Time 4 ± 0.1 (s) Time 5 ± 0.1 (s)

This graph has the raw data of Time (s) and Length of String (m) plotted against one another. The spread of data in the trials increases as the length of string increases from 0.05m to 0.25m; this may be due to the fact that whilst doing the experiment it became more difficult to remain at a constant speed, and therefore a constant string length, the longer the string was. This meant that the data became more spread out. I have set the y axis bounds from 5 seconds to 14 seconds, so that this can be seen more clearly. Processed Data
Table 2

Length of String ± 0.02 (m) 0.05 0.10 0.15 0.20 0.25

Rotations Time 1 Time 2 ± 0.1 (s) ± 0.1 (s) 20 20 20 20 20 6.8 9.1 10.1 11.5 12.6 6.8 8.9 10.6 12.1 13.2

Time 3 Time 4 ± 0.1 (s) ± 0.1 (s) 6.6 8.9 10.2 11.5 12.9 6.7 8.7 10.1 11.4 13.1

Time 5 ± 0.1 (s)

Average time ± 0.1 (s) 6.7 8.9 10.3 11.6 12.9

6.8 9.0 10.3 11.7 12.8

Time Period ± 0.01 (s) 0.34 0.45 0.51 0.58 0.65

Table 2 includes the processed data from my experiment, where I calculated the Average Time (s) using the formula (T₁+T₂+T₃+T₄+T₅)/5 and the Time Period (s) using the formula Average Time (s)/Rotations. Both of these calculations are shown in more detail further on.

3

Table 3

Length of String ± 0.02 (m) 0.05 0.10 0.15 0.20 0.25

Average time ± 0.1 (s) 6.7 8.9 10.3 11.6 12.9

Max time (s) ± 0.1 (s) 6.8 9.1 10.6 12.1 13.2

Min time (s) ± 0.1 (s) 6.6 8.7 10.1 11.4 12.5

Time Period ± 0.01 (s) 0.34 0.45 0.51 0.58 0.65

Time Period² (s²) ± 0.01 (s) 0.11 0.20 0.26 0.34 0.42

Uncertainty Time (s) 0.09 0.20 0.24 0.36 0.35

Uncertainty Time² (s²) 0.01 0.04 0.05 0.13 0.12

Table 3 shows the processed data (Table 2) taken from the original raw data in Table 1. I have also calculated the Time Period² (s²) and Uncertainty Time² (s²), so that these can be plotted on Graph 4.

Processed Data Presentation
Graph 2

Length of String (m) vs Average time (s)
16 14 y = 30.108x + 5.5742

Average Time (s)

12 10 8 6 4 2 0 0 0.05 0.1 0.15 0.2 0.25 0.3 Linear (Average time ± 0.1 (s)) Average time ± 0.1 (s)

Length of String (m)

The non-zero y intercept for Graph 2 is y=5.5742, shown in the equation of the gradient of the graph: y = 30.108x + 5.5742 Graph 2 shows a linear relationship between the Length of String (m) and the Average Time (s). The line of best fit lies within all of the estimated error bars, which implies that the results were precise and accurate. There are no obvious outliers, which reinforces the validity of the data.

4

Graph 3

Length of String (m) vs Time Period (s)
1.2 1 y = 2.5415x + 0.1097

Time Period (s)

0.8 y = 1.5054x + 0.2787 0.6 0.4 0.2 0 0 0.05 0.1 0.15 0.2 0.25 0.3 Time Period ± 0.1 (s) Linear (Time Period ± 0.1 (s)) Linear (Linear Upper Bound) Linear (Linear Lower Bound)

y = 0.5415x + 0.4097

Length of Sring (m)

The non-zero y intercept for this graph is y=0.2787, which was found from the equation of the Linear Time Period line of best fit shown in blue: y = 1.5054x + 0.2787.

Uncertainty in the gradient can be calculated using the formula:

The maximum gradient value for Graph 3 is 2.5415 (shown by the orange trend line), and the minimum gradient is 0.5415 (shown by the green trend line), therefore: ( )

5

Graph 4

Length of String (m) vs Time Period² (s²)
0.6 y = 2.1666x - 0.003 0.5 y = 1.4907x + 0.0424 0.4 0.3 0.2 0.1 0 0 0.05 0.1 0.15 0.2 0.25 0.3 y = 0.8606x + 0.0785 Time Period ± 0.1 (s) Linear (Time Period ± 0.1 (s)) Linear (Linear Upper Bound) Linear (Linear Lower Bound)

Time Period² (s²)

Length of Sring (m)

Graph 4 shows the Length of String (m) vs Time Period² (s²). The gradient of this graph is y= 1.4907, and therefore the uncertainty in the gradient (slope uncertainty) can be calculated using the formula:

The maximum gradient value for Graph 4 is 2.1666 (shown by the orange trend line), and the minimum gradient is 0.8606 (shown by the green trend line), therefore: ( )

Therefore the uncertainty in the gradient is The non-zero y intercept for this graph is y=0.0424, which was found from the equation of the Linear Time Period line of best fit shown in blue: y = 1.4907x + 0.0424.

Calculations for Data Analysis The average time for the processed data was calculated using this formula: (T₁+T₂+T₃+T₄+T₅)/5. The values for time period in Processed Data Tables 2 and 3 was calculated by taking the Average time and dividing by the number of rotations – constant value of 20. For example, the time period when the length of string was 0.05m is: ( ) ( )

I calculated the uncertainty time seen in Table 3 using the formula [MAX Time (s) – MIN Time (s)]/2 for each of the independent variable values; I used these values for the error bars in Graph 3. 6

However the error bars for the raw data in Graph 2 were calculated using the estimated systematic error uncertainty of the stopwatch and factors such as reaction time. From the data I have collected I can now calculate the mass (M) of the stopper. The weight of m is equal to the tension force T=W. The vertical component of the Tension force is equal to the weight of the stopper M. The horizontal force (component) is therefore equal to the centripetal force.

Because we know that

we can say that

and substitute into the equation to get:

Then cancelling out the r we get,

We also know that ( )

therefore,

This formula can then be rearranged to find M in terms of T² and L, which is the gradient of Graph 3.

By substituting real values from the experiment into the equation, the mass of the stopper can be calculated.

To calculate the mass (M) of the stopper, we must first determine the numerical value of W=mg:

By substituting the values of W and T²/L into the equation, you can then solve for M:

7

Absolute Uncertainty of Mass By substituting in the MAX and MIN gradients in Graph 4 into the below equation, the absolute error uncertainty for mass (M) can be calculated.

Therefore the absolute uncertainty in mass (M) is:

(

)

Percentage Error in Mass

Conclusion In conclusion, this experiment was successful because the data I collected was such that I could find the mass of the stopper, within the bounds of the actual mass. From my calculations, I found the mass of the stopper (M) to be 0.010 0.004kg. In reality the actual mass was 0.0127kg, therefore the mass calculated falls within the uncertainty range. I also calculated the percentage uncertainty of M to be 21%, which is close enough to the actual value to state that the results were fairly accurate, however, it is clear that the value of M that I calculated is not the actual value; the reason being mostly related to the weaknesses in the experimental method.

8

From this experiment, I can conclude that the relationship between the time period of the stopper and the length of the string is correlated positively; the longer the string (m) the longer the time period (s). This is shown clearly in Graph 3, with a linear relationship between the two shown by the blue trend line. The linear progression is not perfect, however there were no obvious outliers and the upper and lower bounds of the gradient lie between the error bars in both Graphs 3 and 4. Therefore the results, although not 100% accurate with no random error, are considered accurate enough to allow the actual value of M to lie within the bounds of the calculated mass found using the data in this experiment. Evaluation By looking at the Graph 3, I can see that all of the data points are very close to the line of best-fit, and follow a positive correlation implying that the quality of the data is high. The error bars (although getting larger the longer the length of string) are relatively small, and therefore show the accuracy of the measurements, as the error bars for this graph were calculated from uncertainty time. The longer the length of string, the more spread out the data became: this can be seen in Graph 1. This occurred because whilst doing the experiment, it was much more difficult to remain at a constant frequency the longer the string was and therefore the results varied more due to random error; sometimes the circle of the swinging stopper would be less than horizontal, therefore making the time taken for 20 rotations faster. This was less of a problem for the shorter lengths of string because it was much easier to remain at a constant frequency at these values. There are no obvious outliers in my results, which implies that the data was consistent and that any random error made was not significant enough to affect the overall pattern of the results. The weaknesses in this experiment were mostly due the large human input. The method of recording time periods with a stopwatch would be inaccurate due to human reaction time, and it was very difficult to keep the spinning stopper in a horizontal circle and the frequency constant throughout all of the trials. Also the friction between the string and the edge of the plastic tube was assumed to be negligible in the calculations made, however this would affect the results, meaning the time period I calculated would be shorter than the actual value (that is if all other factors affecting time period were constant). Some of the limitations for this experiment were that the timing had to be done by hand, which introduces a large amount of error into the data. I tried to account for this when I estimated the systematic error to be ± 0.1 (s), however in practise this is likely to be larger. Improving the Investigation To improve this investigation, taking the human element out of the data recording would make the most difference to the results. To do this, apparatus could be set up to spin the stopper at a constant frequency, and sensors put in place to measure exactly how many rotations took place so that when the 20th rotation was complete, the timer would be automatically stopped. This would mean that human reaction time is taken out of the equation, and would therefore give more reliable results. In terms of the friction, a more suitable string or equivalent could be used that creates less friction between the tube and the string, therefore ensuring that less energy is lost as heat to the surroundings. The limitations of this experiment are difficult to overcome if the same method is being used. The only realistic way to improve the limitation regarding time measurement is by acquiring recording equipment that is more precise or in my case doing more trials before taking the average.

9
{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}