The Pendulum

February 6, 2020 

Purpose

To study the factors that might affect the period of a pendulum.

 

Testable Question

Do the mass, amplitude, and length affect the period of a pendulum?

 

Hypothesis/ Prediction/ Background Information

Existing studies^1,2 showed that among the three predictors, the length is positively related to the period of a pendulum. However, the mass and the amplitude are not related to the period of a pendulum. Specifically,  three hypotheses are proposed as follows:  

1. If the length is increased and the mass and amplitude are kept constant, the period will increase.
2. If the mass is increased and amplitude and length are kept fixed, then the period won’t change.
3. If the amplitude is increased and the mass and length are kept constant, then the period won’t change.

 

Variables

Length vs. Period for a Pendulum:

-        Control variables: Mass, amplitude, gravity

-        Independent variable: Length

-        Dependent variables: Period

Mass vs. Period for a Pendulum:

-        Control variables: Length, amplitude, gravity

-        Independent variable: Mass

-        Dependent variables: Period

Amplitude vs. Period for a Pendulum:

-        Control variables: Mass, length, gravity

-        Independent variable: Amplitude

-        Dependent variables: Period

 

Equipment and Materials

-        1 Retort stand

-        1 Clamp

-        1 Ruler

-        1.5 m String

-        1 Tape

-        Hook weight set (20 g, 50 g, 100 g, 200 g, 500 g)

-        1 Stopwatch

-        1 Protractor

 

Procedure

1. Hang 100 g hook weight with a string (1.09 m is the length of string and the bob) on the retort stand clamp.
2. Use the tape to fix the string.  
3. Release the bob from 40° to the vertical equilibrium position. Watch a stopwatch and count the number of full swings at the same time.
4. Stop the stopwatch when the bob completes 10 complete oscillations.
5. Record the total time for 10 cycles.
6. Repeat step 1 to step 5 three times.
7. Repeat step 6 another four times, but change the length to 0.88 m, 0.66 m, 0.46 m, and 0.25 m.
8. Hang 20 g hook weight with a 0.42 m string on the retort stand clamp.
9. Use the tape to fix the string.  
10. Release the bob from 40° to the vertical equilibrium position. Watch a stopwatch and count the number of full swings at the same time.
11. Stop the stopwatch when the bob completes 10 complete oscillations.
12. Record the total time for 10 cycles.
13. Repeat step 10 to step 12 three times.
14. Repeat step 13 another four times but change the mass of bob to 50 g, 100 g, 200 g, and 500 g.
15. Hang 100 g hook weight with a string (0.42 m is the length of string and the bob) on the retort stand clamp.
16. Use the tape to fix the string.
17. Release the bob from 45° to the vertical equilibrium position. Watch a stopwatch and count the number of full swings at the same time.
18. Stop the stopwatch when the bob completes 10 complete oscillations.
19. Record the total time for 10 cycles.
20. Repeat step 17 to step 19 three times.
21. Repeat step 20 another four times but change the amplitude to 40°, 30°, 20°, and 10°.

 

Quantitative Observations

 

Table 1: Length vs. Period for a Pendulum

Length(m)	Mass (g)	Amplitude (˚)	Total Time for 10 cycles (s)			Average Time for 10 cycles (s)	Average Period, T (s)
1.09 m	100 g	40˚	21.65 s	21.56 s	21.30 s	21.50 s	2.150 s
0.88 m	100 g	40˚	19.33 s	19.38 s	19.22 s	19.31 s	1.931 s
0.66 m	100 g	40˚	16.52 s	16.73 s	16.72 s	16.66 s	1.666 s
0.46 m	100 g	40˚	13.68 s	13.75 s	13.90 s	13.78 s	1.378 s
0.25 m	100 g	40˚	10.32 s	10.26 s	10.36 s	10.31 s	1.031 s

Note, Length (m) refers to the length of the string and the length of the bob.

 

Table 2: Mass vs. Period for a Pendulum

Length(m)	Mass (g)	Amplitude (˚)	Total Time for 10 cycles (s)			Average Time for 10 cycles (s)	Average Period, T (s)
0.42 m	20 g	40˚	13.59 s	13.65 s	13.53 s	13.59 s	1.359 s
0.42 m	50 g	40˚	13.78 s	13.71 s	13.72 s	13.74 s	1.374 s
0.42 m	100 g	40˚	13.95 s	14.03 s	13.89 s	13.96 s	1.396 s
0.42 m	200 g	40˚	13.85 s	13.75 s	13.96 s	13.85 s	1.385 s
0.42 m	500 g	40˚	14.12 s	13.93 s	14.12 s	14.06 s	1.406 s

Note, Length (m) refers to the length of the string.

 

 Table 3: Amplitude vs. Period for a Pendulum

Length(m)	Mass (g)	Amplitude (˚)	Total Time for 10 cycles (s)			Average Time for 10 cycles (s)	Average Period, T (s)
0.42 m	100 g	45˚	12.861 s	12.861 s	12.861 s	12.861 s	1.2861 s
0.42 m	100 g	40˚	12.754 s	12.754 s	12.754 s	12.754 s	1.2754 s
0.42 m	100 g	30˚	12.582 s	12.582 s	12.582 s	12.582 s	1.2582 s
0.42 m	100 g	20˚	12.461 s	12.461 s	12.461 s	12.461 s	1.2461 s
0.42 m	100 g	10˚	12.390 s	12.390 s	12.390 s	12.390 s	1.2390 s

Note, Length (m) refers to the length of the string and the length of the bob.

 

Qualitative Observations

1. The bob accidentally hit the counter while it was oscillating.
2. One of the group members held the retort stand while using the 500 g bob in doing the lab because the retort stand was shaking.

 

Analysis

The square of the period is proportional to the length, and they have a linear relationship.

As the mass increases, the period remains the same. There is no relationship between the two variables (mass and period).

As the amplitude increases, the period remains the same. There is no relationship between the two variables (amplitude and period).

 

Evaluation 

The hypothesis that was made before the lab was supported by the results of the analysis. As expected, when the mass is increased and amplitude and length are kept fixed, the period doesn’t change the value. When the amplitude is increased and the mass and length are kept constant, the period doesn’t change.  Hypotheses 2 and 3 are both supported. 

However, the relationship between length and period is not a linear relationship. Their relationship is demonstrated in terms of a positive relationship between length and the square of the period. Namely, the square of the period is directly proportional to the length when the mass and the amplitude are kept constant. This result is not completely the same as hypothesis 1.

In addition to the aforementioned quantitative results, qualitative observations highlight two possible errors might affect the accuracy of data collected in this lab. In the procedure, the bob accidentally hit the counter during it was oscillating and this would lead to bias in the time of the oscillation. The second error was that our group decided to add an additional procedure. One of the group members held the retort stand while using the heaviest bob in doing the lab because the retort stand was shaking. This unexpected procedure might also lead to bias in the measurement. The two efforts may cause slight differences in the values which were shown on the lines in Figures 2 and 3.

 

Conclusion

        Based on the evaluation, only length affects the period of a pendulum. Mass and amplitude don’t affect the period of a pendulum.

 

Citations

Citations 1: https://www.physicsclassroom.com/class/waves/Lesson-0/Pendulum-Motion

Citations 2: https://baike.baidu.com/item/%E5%8D%95%E6%91%86 

Acceleration of Gravity

March 13th, 2020

Purpose

The purpose of this lab is to prove the acceleration due to gravity on earth is 9.81 m/s² down.  Therefore, this will be examined by analyzing the motion path of a free fall eraser.

 

Testable Question

        Does the eraser go through uniform acceleration while free falling?

Hypothesis/ Prediction/ Background Information

        The acceleration due to gravity that the eraser experiences will be constant, which is 9.81 m/s² down because gravity on earth causes the eraser to accelerate toward the centre of the earth.

Variables

Control variables: Mass

Independent variable: Time

Dependent variables: Velocity

Equipment and Materials

-        1 Meter stick

-        1 Eraser

-        1 Smartphone to record video

-        Tracker software

Procedure

1. Place the meter stick in the frame of the camera.
2. Recording the motion path of the eraser while dropping the eraser straight down without initial velocity.
3. Use Tracker to produce the velocity-time graph and table of the eraser’s motion.
4. Obtain the values of velocity at every certain time interval from the table that is created by Tracker software and record them in the observation table.
5. Record down the slopes of each certain time interval from the velocity-time graph in the observation table.
6. Analyze the acceleration due to gravity and initial velocity from the equation which is vy = A × t + B.

Quantitative Observations

Table: Time, Velocity, Acceleration for the Free Fall Eraser

Time (s)	Velocity (m/s)	Acceleration (m/s2)
0.000
3.333E-2	-0.300
6.667E-2	-0.675	-1.12E1
0.100	-1.050	-1.12E1
0.133	-1.423	-9.54E0
0.167	-1.686	-8.44E0
0.200	-1.985	-1.13E1
0.233	-2.439	-1.22E1
0.267	-2.798	-6.24E0
0.300	-2.855	-6.81E0
0.333	-3.252
0.367

*Let up to be positive.

Qualitative Observations

The initial velocity is not absolutely zero. Before dropping down the eraser, there was some initial velocity down.

 

    Figure 1: Before dropping the eraser    Figure 2: The motion path of the eraser

Analysis

Figure 3: Time vs. Velocity of the Free Fall Eraser

*Let up to be positive.

 

*vy = A × t + B:

A is the slope which is the acceleration due to gravity, and B is the y-intercept of the graph which is the initial velocity of the eraser.

As time increases, the value of the velocity decreases. But, the negative sign on the graph means that the direction is going down. Therefore, as time increases, the absolute value of the velocity also increases (which means that as time increases, the velocity is faster). Furthermore, there is a linear relationship between time and velocity, and the correlation between time and the absolute value of velocity is positive.

Percentage Error: 

|(Theoretical - Experimental)÷Theoretical|×100%

=|(-9.806-(-9.804))÷(-9.806)|×100% 

= 0.02040%

 

Evaluation 

The hypothesis that was made before the lab is not completely supported by the result. From the slope of each of the points, not every point of time has the same acceleration, which means that the acceleration is not constant. However, from the best fit line of the velocity-time graph, the slope which is the acceleration of the free fall eraser’s whole motion is -9.804E0 m/s². That value is very close to the value of 9.81 m/s² down. Therefore, the differences between the accelerations due to gravity in every point of time and the hypothesized value (i.e. 9.81 m/s² down) may be caused by the errors. The sources of these errors can come from the initial velocity of the eraser, the air resistance, or the tracking points which were not exactly the same as the eraser actually did.

 

Conclusion

While an object is free-falling, the velocity increases along with the increase in time. The slope of the best fit line that explains the variances of time and velocity is 9.81 m/s2 down. However, errors actually exist, and that’s why there are some distances from data points to the best line. 

 

Citations

Acceleration due to gravity. (2020, January 23). Retrieved March 12, 2020, from https://simple.wikipedia.org/wiki/Acceleration_due_to_gravity