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Posted: February 9th, 2023

# Simple Harmonic Motion

Project 2
Evaluation 32
Physics 2 (SCIH 036 058)

Simple Harmonic Motion
Introduction
Both light and sound are dependent on waves. Sound is communicated via pressure waves in air and other materials while light is communicated via electromagnetic waves. Both light and sound, are examples of simple harmonic motion. In this project you will construct and use a simple pendulum (a mass on a string) to illustrate the principles of simple harmonic motion.
You will need the following materials to complete this project. The balance and protractor can be found in your Lab Kit. You will be responsible for providing the other necessary materials.
Mass (Various materials can be used. You can use small household items such as metal
washers, a small bag of sand, a rock, a 9-V battery, or any other small item that has a
significant mass for its size and can be attached to the end of the string.)
String (lightweight, strong – like dental floss)
Protractor
Scale or balance
Timer / Stopwatch / Watch with a second hand
Meter stick (or equivalent)

Procedure
Part A: Relationship between Mass and Period
(25 points possible)
1. Using some string and a mass, build a simple pendulum. The higher your pendulum is, the longer your string is and the easier it will be to measure (and more accurate). (A pendulum is loosely defined as something hanging from a fixed point which, when pulled back and released, is free to swing down by gravity and then out and up because of its inertia, or tendency to stay in motion.)

2. Measure your mass, length of string, and planned amplitude carefully. Remember that the amplitude is the distance the mass is pulled from the centerline.
Length of String (from pivot to center of mass), in cm: 120cm
Amplitude (from centerline to farthest distance), in cm: 20cm

The “period” of a pendulum is the amount of time it takes the mass to complete one full cycle. Pull the mass back 10 or 20 centimeters from the starting point; let it swing out to the other side and back to its original starting point. The time it takes to complete that entire path is the period. Because this happens very quickly, it is a good idea to measure the time it takes to complete 10 cycles and then divide by 10 to get the time for one cycle.

Prediction: What do you think will happen to the period as you change the mass? Record your prediction here:
As mass increases the period will be shorter.
3. When you are ready to begin, pull the mass back to the desired amplitude and let the pendulum start swinging.
4. Let the pendulum complete one full cycle before you begin timing.
5. Using a stopwatch (or your watch with a second hand), measure the time it takes to complete 10 full cycles. Repeat this approximately 6 times. Calculate the period by dividing each time by 10 and then find the average period.
Mass (g): 30g
Time for 10 full cycles (s) Period (s)
Rep 1 22.1 2.21
Rep 2 22.1 2.21
Rep 3 22.2 2.22
Rep 4 22.2 2.22
Rep 5 22.1 2.21
Rep 6 22.1 2.21
Average (s): 2.21

6. Repeat this procedure with at least two additional masses. Be sure to keep your pendulum string the same length and to pull the mass back to the same amplitude each time. Record your results in the tables that follow.

Mass (g): 60g
Time for 10 full cycles (s)
Period (s)
Rep 1 22.2 2.22
Rep 2 22.3 2.23
Rep 3 22.2 2.22
Rep 4 22.3 2.23
Rep 5 22.2 2.22
Rep 6 22.3 2.23
Average (s): 2.23

Mass (g): 90g
Time for 10 full cycles (s) Period (s)
Rep 1 22.7 2.27
22.6 22.6 2.26
Rep 3 22.6 2.26
Rep 4 22.6 2.26
Rep 5 22.6 2.26
Rep 6 22.6 2.26
Average (in seconds): 2.26

7. Record your overall results in the following table.
Mass (g) 30 Period (s) 2.21
Mass (g) 60 Period (s) 2.23
Mass (g) 90 Period (s) 2.26

8. Graph your results with mass as the independent variable (x-axis) and period as the dependent variable (y-axis). Although graphing by hand is acceptable, it is often helpful to use a graphing program like Excel. If you choose to graph on your computer and then import it into this project, be sure to use the following settings:

• Use a scatterplot with no lines.
• Adjust scale to start from zero and show all the data.
o -Do NOT force the trendline to go through the origin (0,0).
o -DO add the equation of the line to the graph.
o -DO add the r2 value to the graph (If the r2 value is greater than 0.9 you have a reasonably linear graph. If it is less than 0.9 you may want to think about collecting your data again.).

If you choose to make the graph by hand, you can use the following or your own graph paper:

9. Determine the slope of the line and the general equation of the line (use the form: y = mx + b)

10. What does the slope mean?

13. What is the relationship between mass and period?

Part B: Relationship between Amplitude and Period
(25 points possible)
Amplitude is the distance that the mass is pulled back from the centerline. Amplitude is related to the angle of the string and we could measure the angle but measuring amplitude is easier. Simply lay a meter-stick under your pendulum and note how far back you are pulling the mass. Be sure to start with a fairly small amplitude. Be sure to keep your mass and string length constant.

Prediction: What do you think will happen to the period as you change the amplitude? Record your prediction here:
The period will get longer as you increase the amplitude.

14. Starting with a small amplitude, measure the time it takes the pendulum to complete 10 cycles. Collect data for at least 3 different amplitudes. As before, use this data to determine the period of the pendulum at this amplitude.
Amplitude 1 (cm) 10
Mass (g) 90g
Length (cm) 120cm
Time for 10 full cycles (s)
Period (s)
Rep 1 22.5 2.25
Rep 2 22.5 2.25
Rep 3 22.5 2.25
Rep 4 22.4 2.24
Rep 5 22.5 2.25
Rep 6 22.4 2.24
Average (s) 2.25

Amplitude 2 (cm) 20cm
Mass (g) 90g
Length (cm) 120cm
Time for 10 full cycles (s) Period (s)
Rep 1 22.6 2.26
Rep 2 22.6 2.26
Rep 3 22.6 2.26
Rep 4 22.7 2.27
Rep 5 22.6 2.26
Rep 6 22.7 2.27
Average (s) 2.26

Amplitude 3 (cm) 30cm
Mass (g) 90g
Length (cm) 120cm
Time for 10 full cycles (s) Period (s)
Rep 1 22.7 2.27
Rep 2 22.7 2.27
Rep 3 22.7 2.27
Rep 4 22.8 2.28
Rep 5 22.7 2.27
Rep 6 22.8 2.28
Average (s) 2.27

15. Record your overall results in the following table.
Amplitude 1 (cm) 10 Period (s) 2.25
Amplitude 2 (cm) 20 Period (s) 2.26
Amplitude 3 (cm) 30 Period (s) 2.27

16. Graph your results with amplitude as the independent variable (x-axis) and period as the dependent variable (y-axis).

17. Determine the slope of the line and the general equation of the line (use the form: y = mx + b)

18. What does the slope mean?

21. What is the relationship between amplitude and period?

Part C: Relationship between Length and Period
(25 points possible)
You have kept the length of the string constant to this point. In this step you will choose 3 different lengths to test. Be sure to keep mass and amplitude constant.
Prediction: What do you think will happen to the period as you change the length? Record your prediction here:

22. Starting with a small length, measure the time it takes the pendulum to complete 10 cycles. Collect data for at least 3 different lengths. As before, use this data to determine the period of the pendulum at this amplitude.
Length 1 (cm) 40cm
Mass (g) 90g
Amplitude (cm) 30cm
Time for 10 full cycles (s)
Period (s)
Rep 1 13.6 1.36
Rep 2 13.7 1.37
Rep 3 13.7 1.37
Rep 4 13.7 1.37
Rep 5 13.6 1.36
Rep 6 13.7 1.37
Average (s) 1.37

Length 2 (cm) 80cm
Mass (g) 90g
Amplitude (in cm): 30cm
Time for 10 full cycles (s) Period (s)
Rep 1 18.9 1.89
Rep 2 18.8 1.88
Rep 3 18.8 1.88
Rep 4 18.9 1.89
Rep 5 18.9 1.89
Rep 6 18.9 1.89
Average (s) 1.89

Length 3 (cm) 120cm
Mass (g) 90g
Amplitude (cm) 30cm
Time for 10 full cycles (s) Period (s)
Rep 1 22.8 2.28
Rep 2 22.7 2.27
Rep 3 22.7 2.27
Rep 4 22.7 2.27
Rep 5 22.7 2.27
Rep 6 22.8 2.28
Average (s) 2.27

23. Record your overall results in the table below.
Length 1 (cm) 40cm Period (s) 1.37
Length 2 (cm) 80cm Period (s) 1.89
Length 3 (cm) 120cm Period (s) 2.27

24. Graph your results with Length as the independent variable (x-axis) and Period as the dependent variable (y-axis).

25. Determine the slope of the line and the general equation of the line (use the form: y = mx + b)

26. What does the slope mean?

29. What is the relationship between length and period?

Part D: Evaluation of Data
(25 points possible)
The period of a simple pendulum is predicted by the equation where
T = period
L = length of pendulum from pivot to center of mass
g = acceleration due to gravity (9.8m/s2)
Compare your data to the period predicted by this formula.

30. How well do they compare?

31. List some factors that might contribute to a difference between your data and the ideal period that was predicted.

32. Determine the period of your longest pendulum if it were on the moon where the acceleration due to gravity is only 1.6m/s2.

33. In order to swing with a period of exactly 2.0 s, a grandfather clock’s 1.5 kg pendulum must have a length of ______m?

34. Write a summary (100-200 words) that discusses
a. the relationships you found between mass, amplitude, length, and period (Be sure your discussion is based on YOUR data.).
b. how closely your results compare to the results predicted by the pendulum equation.
c. the errors / approximations that were most likely to have affected your results.

Objective Exceeds minimum project expectations Meets minimum project expectations Approaches course expectations Does not meet course expectations
Construction and Data Charts and graphs completed accurately. Data is easy to read and comprehend. Charts and graphs represent the minimum requirements but may be difficult to read (graph too small, or elements not clearly identified). Some charts or graphs do not accurately reflect data required. Some parts completed, but some lacking in definition or specificity. Construction of charts and graphs does not reflect instructions.
Calculations All calculations completed accurately. All work is shown and calculations are easy to follow. Most calculations completed accurately. Most of the work is shown and calculations are somewhat easy to follow. Some calculations completed accurately. Not all work is shown and some calculations are difficult to follow. Calculations not completed accurately. Not all work is shown.
Evaluation of Data All explanations are clear, concise, and accurate for the data presented. Questions are answered thoroughly and with evident reflection and understanding of the concepts. Most explanations are clear, concise, and accurate for the data presented. Questions were answered but could have shown a more thorough understanding of the concepts. Some explanations are clear and accurate for the data presented. Student does not demonstrate a good understanding of the concepts. Evaluation of data is not complete.
Possible Grade (in percentage points) 90-100 80-90 70-80 69 or below

This project can be submitted electronically. Check the Project page under “My Work” in the ISHS online course management system or your enrollment information with your print materials for more detailed instructions.

===>
Introduction
Simple harmonic motion is a concept that is important in both light and sound. Sound is conveyed through pressure waves in air and other materials, while light is communicated through electromagnetic waves. In this project, the simple harmonic motion was demonstrated using a simple pendulum (a mass on a string).

Materials and Procedure
For this project, a mass (various materials can be used such as metal washers, sand, rocks, a 9-V battery, or any other small item with a significant mass), string (lightweight and strong, like dental floss), protractor, scale or balance, timer or stopwatch, and meter stick were used. The length of the string (from pivot to center of mass) was 120 cm and the amplitude (from centerline to farthest distance) was 20 cm.

The “period” of a pendulum is defined as the amount of time it takes for the mass to complete one full cycle. The pendulum was pulled back 10 or 20 cm from the starting point, then let it swing out to the other side and back to its original starting point. The time it took to complete the entire path was recorded as the period. The period was calculated by measuring the time it took to complete 10 cycles, dividing by 10 to get the time for one cycle.

The period was measured for three different masses (30g, 60g, and 90g) and the results were recorded. The average period for each mass was calculated. The results were then recorded in a table and graphed with mass as the independent variable (x-axis) and period as the dependent variable (y-axis).

Results and Conclusion
The results showed that as the mass increased, the period decreased. The overall results were recorded in a table and graphed to show the relationship between mass and period. The graph confirmed that as mass increased, the period decreased.

This project demonstrated the principles of simple harmonic motion using a simple pendulum. It also showed the relationship between mass and period in a simple pendulum. These results can be used to understand more complex wave interactions and phenomena in the future.
===>
Project 2
Evaluation 32
Physics 2 (SCIH 036 058)

Simple Harmonic Motion
Introduction
Both light and sound are dependent on waves. Sound is communicated via pressure waves in air and other materials while light is communicated via electromagnetic waves. Both light and sound, are examples of simple harmonic motion. In this project you will construct and use a simple pendulum (a mass on a string) to illustrate the principles of simple harmonic motion.
You will need the following materials to complete this project. The balance and protractor can be found in your Lab Kit. You will be responsible for providing the other necessary materials.
Mass (Various materials can be used. You can use small household items such as metal
washers, a small bag of sand, a rock, a 9-V battery, or any other small item that has a
significant mass for its size and can be attached to the end of the string.)
String (lightweight, strong – like dental floss)
Protractor
Scale or balance
Timer / Stopwatch / Watch with a second hand
Meter stick (or equivalent)

Procedure
Part A: Relationship between Mass and Period
(25 points possible)
1. Using some string and a mass, build a simple pendulum. The higher your pendulum is, the longer your string is and the easier it will be to measure (and more accurate). (A pendulum is loosely defined as something hanging from a fixed point which, when pulled back and released, is free to swing down by gravity and then out and up because of its inertia, or tendency to stay in motion.)

2. Measure your mass, length of string, and planned amplitude carefully. Remember that the amplitude is the distance the mass is pulled from the centerline.
Length of String (from pivot to center of mass), in cm: 120cm
Amplitude (from centerline to farthest distance), in cm: 20cm

The “period” of a pendulum is the amount of time it takes the mass to complete one full cycle. Pull the mass back 10 or 20 centimeters from the starting point; let it swing out to the other side and back to its original starting point. The time it takes to complete that entire path is the period. Because this happens very quickly, it is a good idea to measure the time it takes to complete 10 cycles and then divide by 10 to get the time for one cycle.

Prediction: What do you think will happen to the period as you change the mass? Record your prediction here:
As mass increases the period will be shorter.
3. When you are ready to begin, pull the mass back to the desired amplitude and let the pendulum start swinging.
4. Let the pendulum complete one full cycle before you begin timing.
5. Using a stopwatch (or your watch with a second hand), measure the time it takes to complete 10 full cycles. Repeat this approximately 6 times. Calculate the period by dividing each time by 10 and then find the average period.
Mass (g): 30g
Time for 10 full cycles (s) Period (s)
Rep 1 22.1 2.21
Rep 2 22.1 2.21
Rep 3 22.2 2.22
Rep 4 22.2 2.22
Rep 5 22.1 2.21
Rep 6 22.1 2.21
Average (s): 2.21

6. Repeat this procedure with at least two additional masses. Be sure to keep your pendulum string the same length and to pull the mass back to the same amplitude each time. Record your results in the tables that follow.

Mass (g): 60g
Time for 10 full cycles (s)
Period (s)
Rep 1 22.2 2.22
Rep 2 22.3 2.23
Rep 3 22.2 2.22
Rep 4 22.3 2.23
Rep 5 22.2 2.22
Rep 6 22.3 2.23
Average (s): 2.23

Mass (g): 90g
Time for 10 full cycles (s) Period (s)
Rep 1 22.7 2.27
22.6 22.6 2.26
Rep 3 22.6 2.26
Rep 4 22.6 2.26
Rep 5 22.6 2.26
Rep 6 22.6 2.26
Average (in seconds): 2.26

7. Record your overall results in the following table.
Mass (g) 30 Period (s) 2.21
Mass (g) 60 Period (s) 2.23
Mass (g) 90 Period (s) 2.26

8. Graph your results with mass as the independent variable (x-axis) and period as the dependent variable (y-axis). Although graphing by hand is acceptable, it is often helpful to use a graphing program like Excel. If you choose to graph on your computer and then import it into this project, be sure to use the following settings:

• Use a scatterplot with no lines.
• Adjust scale to start from zero and show all the data.
o -Do NOT force the trendline to go through the origin (0,0).
o -DO add the equation of the line to the graph.
o -DO add the r2 value to the graph (If the r2 value is greater than 0.9 you have a reasonably linear graph. If it is less than 0.9 you may want to think about collecting your data again.).

If you choose to make the graph by hand, you can use the following or your own graph paper:

9. Determine the slope of the line and the general equation of the line (use the form: y = mx + b)

10. What does the slope mean?

13. What is the relationship between mass and period?

Part B: Relationship between Amplitude and Period
(25 points possible)
Amplitude is the distance that the mass is pulled back from the centerline. Amplitude is related to the angle of the string and we could measure the angle but measuring amplitude is easier. Simply lay a meter-stick under your pendulum and note how far back you are pulling the mass. Be sure to start with a fairly small amplitude. Be sure to keep your mass and string length constant.

Prediction: What do you think will happen to the period as you change the amplitude? Record your prediction here:
The period will get longer as you increase the amplitude.

14. Starting with a small amplitude, measure the time it takes the pendulum to complete 10 cycles. Collect data for at least 3 different amplitudes. As before, use this data to determine the period of the pendulum at this amplitude.
Amplitude 1 (cm) 10
Mass (g) 90g
Length (cm) 120cm
Time for 10 full cycles (s)
Period (s)
Rep 1 22.5 2.25
Rep 2 22.5 2.25
Rep 3 22.5 2.25
Rep 4 22.4 2.24
Rep 5 22.5 2.25
Rep 6 22.4 2.24
Average (s) 2.25

Amplitude 2 (cm) 20cm
Mass (g) 90g
Length (cm) 120cm
Time for 10 full cycles (s) Period (s)
Rep 1 22.6 2.26
Rep 2 22.6 2.26
Rep 3 22.6 2.26
Rep 4 22.7 2.27
Rep 5 22.6 2.26
Rep 6 22.7 2.27
Average (s) 2.26

Amplitude 3 (cm) 30cm
Mass (g) 90g
Length (cm) 120cm
Time for 10 full cycles (s) Period (s)
Rep 1 22.7 2.27
Rep 2 22.7 2.27
Rep 3 22.7 2.27
Rep 4 22.8 2.28
Rep 5 22.7 2.27
Rep 6 22.8 2.28
Average (s) 2.27

15. Record your overall results in the following table.
Amplitude 1 (cm) 10 Period (s) 2.25
Amplitude 2 (cm) 20 Period (s) 2.26
Amplitude 3 (cm) 30 Period (s) 2.27

16. Graph your results with amplitude as the independent variable (x-axis) and period as the dependent variable (y-axis).

17. Determine the slope of the line and the general equation of the line (use the form: y = mx + b)

18. What does the slope mean?

21. What is the relationship between amplitude and period?

Part C: Relationship between Length and Period
(25 points possible)
You have kept the length of the string constant to this point. In this step you will choose 3 different lengths to test. Be sure to keep mass and amplitude constant.
Prediction: What do you think will happen to the period as you change the length? Record your prediction here:

22. Starting with a small length, measure the time it takes the pendulum to complete 10 cycles. Collect data for at least 3 different lengths. As before, use this data to determine the period of the pendulum at this amplitude.
Length 1 (cm) 40cm
Mass (g) 90g
Amplitude (cm) 30cm
Time for 10 full cycles (s)
Period (s)
Rep 1 13.6 1.36
Rep 2 13.7 1.37
Rep 3 13.7 1.37
Rep 4 13.7 1.37
Rep 5 13.6 1.36
Rep 6 13.7 1.37
Average (s) 1.37

Length 2 (cm) 80cm
Mass (g) 90g
Amplitude (in cm): 30cm
Time for 10 full cycles (s) Period (s)
Rep 1 18.9 1.89
Rep 2 18.8 1.88
Rep 3 18.8 1.88
Rep 4 18.9 1.89
Rep 5 18.9 1.89
Rep 6 18.9 1.89
Average (s) 1.89

Length 3 (cm) 120cm
Mass (g) 90g
Amplitude (cm) 30cm
Time for 10 full cycles (s) Period (s)
Rep 1 22.8 2.28
Rep 2 22.7 2.27
Rep 3 22.7 2.27
Rep 4 22.7 2.27
Rep 5 22.7 2.27
Rep 6 22.8 2.28
Average (s) 2.27

23. Record your overall results in the table below.
Length 1 (cm) 40cm Period (s) 1.37
Length 2 (cm) 80cm Period (s) 1.89
Length 3 (cm) 120cm Period (s) 2.27

24. Graph your results with Length as the independent variable (x-axis) and Period as the dependent variable (y-axis).

25. Determine the slope of the line and the general equation of the line (use the form: y = mx + b)

26. What does the slope mean?

29. What is the relationship between length and period?

Part D: Evaluation of Data
(25 points possible)
The period of a simple pendulum is predicted by the equation where
T = period
L = length of pendulum from pivot to center of mass
g = acceleration due to gravity (9.8m/s2)
Compare your data to the period predicted by this formula.

30. How well do they compare?

31. List some factors that might contribute to a difference between your data and the ideal period that was predicted.

32. Determine the period of your longest pendulum if it were on the moon where the acceleration due to gravity is only 1.6m/s2.

33. In order to swing with a period of exactly 2.0 s, a grandfather clock’s 1.5 kg pendulum must have a length of ______m?

34. Write a summary (100-200 words) that discusses
a. the relationships you found between mass, amplitude, length, and period (Be sure your discussion is based on YOUR data.).
b. how closely your results compare to the results predicted by the pendulum equation.
c. the errors / approximations that were most likely to have affected your results.

Objective Exceeds minimum project expectations Meets minimum project expectations Approaches course expectations Does not meet course expectations
Construction and Data Charts and graphs completed accurately. Data is easy to read and comprehend. Charts and graphs represent the minimum requirements but may be difficult to read (graph too small, or elements not clearly identified). Some charts or graphs do not accurately reflect data required. Some parts completed, but some lacking in definition or specificity. Construction of charts and graphs does not reflect instructions.
Calculations All calculations completed accurately. All work is shown and calculations are easy to follow. Most calculations completed accurately. Most of the work is shown and calculations are somewhat easy to follow. Some calculations completed accurately. Not all work is shown and some calculations are difficult to follow. Calculations not completed accurately. Not all work is shown.
Evaluation of Data All explanations are clear, concise, and accurate for the data presented. Questions are answered thoroughly and with evident reflection and understanding of the concepts. Most explanations are clear, concise, and accurate for the data presented. Questions were answered but could have shown a more thorough understanding of the concepts. Some explanations are clear and accurate for the data presented. Student does not demonstrate a good understanding of the concepts. Evaluation of data is not complete.
Possible Grade (in percentage points) 90-100 80-90 70-80 69 or below

This project can be submitted electronically. Check the Project page under “My Work” in the ISHS online course management system or your enrollment information with your print materials for more detailed instructions.
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Name: _________________________ ID: _______________________
Project 2: SIMPLE HARMONIC MOTION (SHM)
Evaluation 32
Course: Physics 2 (SCIH 036 058)
Title: Illustrating the Principles of Simple Harmonic Motion (SHM)
Objective: To construct and utilize a pendulum to demonstrate the principles of the SHM
Introduction: Most objects oscillate or vibrate in a periodic pattern. The vibrations develop systematic motions known as waves. Sound and light are two phenomena that depend on waves for their transmission. Pressure waves aid to sound transmission whereas electromagnetic waves aid to transmission of light. However, both light and sound oscillate in a simple harmonic motion. The principles entailed in simple harmonic motion can be illustrated using an experiment with a pendulum suspension to measure the movement properties on changes on suspension length ‘l’, amplitude ‘A’, and period ‘T’, among others. During the evaluations, we take g=9.8m/s2. Period, T is dependent on length, l and gravity due to earth, g according to the equation;
T=2π√((l)/g)
We can rearrange to get g= (4 π2 l/T2)
Materials: Balance, protractor, strong light string, stop watch, meter stick, mass (30g, 60g and 90g).
Methodology: Part A- How mass (m) relates to period (T)
Build a simple pendulum using separate masses (30g, 60g and 90g) and a string. The longer the length, l, of a pendulum the better in terms of accurate measures.
Record the amplitude, and the pendulum length. The amplitude will be used for all the masses during the experiment.
Release the pendulum from the estimated amplitude and measure the time taken for 10 cycles. Divide the total time taken for 10 oscillations by 10 to get the period. It is advised to let the pendulum oscillate once before beginning your counts.
Repeat the steps above for six trials.
Find the period, T for each trial at the same length and amplitude.
Draw a graph of mass, m (x-axis) as an independent variable and period, T (y-axis) as a dependent variable to determine the relationship between mass and pendulum length.
Prediction: As mass increases, the period becomes shorter.
Results and Analysis:
Mass (g): 30g
Time for 10 full cycles (s) Period (s)
Rep 1 22.1 2.21
Rep 2 22.1 2.21
Rep 3 22.2 2.22
Rep 4 22.2 2.22
Rep 5 22.1 2.21
Rep 6 22.1 2.21
Average (s): 2.21
Mass (g): 60g
Time for 10 full cycles (s)
Period (s)
Rep 1 22.2 2.22
Rep 2 22.3 2.23
Rep 3 22.2 2.22
Rep 4 22.3 2.23
Rep 5 22.2 2.22
Rep 6 22.3 2.23
Average (s): 2.23
Mass (g): 90g
Time for 10 full cycles (s) Period (s)
Rep 1 22.7 2.27
22.6 22.6 2.26
Rep 3 22.6 2.26
Rep 4 22.6 2.26
Rep 5 22.6 2.26
Rep 6 22.6 2.26
Average (in seconds): 2.26
Overall Result
Mass (g) 30 Period (s) 2.21
Mass (g) 60 Period (s) 2.23
Mass (g) 90 Period (s) 2.26

Graphical Analysis
Graph of mass (m) against period (T)

Slope of the curve: Slope=Δy/Δx= (2.25-2.2)/(80-20)
slope=(〖8.3×10〗^(-4) s)/g or 0.83 s/kg
To form an equation, we need to know the y-intercept. At y-intercept, x=0. With the gradient already known, we can get the y-intercept as follows.
〖8.3×10〗^(-4)=(2.2-y)/(20-0)
Therefore, y-intercept is at y=2.1833
Hence, the equation of the line will be: 〖y=8.3×10〗^(-4) x+2.18
The slope implies that period (T) in the SHM increases with increase in mass (m) which is different from my initial prediction.
Methodology: Part B- How amplitude (A) relates to period (T)
Keep the pendulum length (120cm) and mass (90g) constant, vary the oscillation amplitudes by 10cm, 20cm, and 30cm, and measure the respective oscillation periods by averaging periods in six different trials per test.
The plot a graph of amplitude (independent variable, x-axis) against period (dependent variable, y-axis)
Prediction: The longer the amplitude, the longer the period.
Results and Analysis
Amplitude 1 (cm) 10
Mass (g) 90g
Length (cm) 120cm
Time for 10 full cycles (s)
Period (s)
Rep 1 22.5 2.25
Rep 2 22.5 2.25
Rep 3 22.5 2.25
Rep 4 22.4 2.24
Rep 5 22.5 2.25
Rep 6 22.4 2.24
Average (s) 2.25

Amplitude 2 (cm) 20cm
Mass (g) 90g
Length (cm) 120cm
Time for 10 full cycles (s) Period (s)
Rep 1 22.6 2.26
Rep 2 22.6 2.26
Rep 3 22.6 2.26
Rep 4 22.7 2.27
Rep 5 22.6 2.26
Rep 6 22.7 2.27
Average (s) 2.26
Amplitude 3 (cm) 30cm
Mass (g) 90g
Length (cm) 120cm
Time for 10 full cycles (s) Period (s)
Rep 1 22.7 2.27
Rep 2 22.7 2.27
Rep 3 22.7 2.27
Rep 4 22.8 2.28
Rep 5 22.7 2.27
Rep 6 22.8 2.28
Average (s) 2.27
Overall Results
Amplitude 1 (cm) 10 Period (s) 2.25
Amplitude 2 (cm) 20 Period (s) 2.26
Amplitude 3 (cm) 30 Period (s) 2.27

Graphical Analysis
Graph of amplitude (A) in cm against period (T) in s

Slope of the curve: Slope=Δy/Δx= (2.28-2.24)/(40-0)
slope=(〖1.0×10〗^(-3) s)/cm or 0.1 s/m
To form an equation, we need to know the y-intercept. At y-intercept, x=0. With the gradient already known, we can get the y-intercept as follows.
〖1.0×10〗^(-3)=(2.28-y)/(40-0)
Therefore, y-intercept is at y=2.24
Hence, the equation of the line will be: 〖y=1.0×10〗^(-3) x+2.24
The slope implies that period (T) in the SHM increases with increase in amplitude (A).
This finding proved my initial prediction.
Therefore, period is directly proportional to amplitude in a SHM.
Methodology: Part C- How length (l) relates to period (T)
In this step, vary the pendulum lengths (40cm, 80cm, and 120cm) as the period and amplitude remain constant. Collect the data three times using six trials per test.
Prediction: The shorter the pendulum length, the shorter the period.
Results and Analysis
Length 1 (cm) 40cm
Mass (g) 90g
Amplitude (cm) 30cm
Time for 10 full cycles (s)
Period (s)
Rep 1 13.6 1.36
Rep 2 13.7 1.37
Rep 3 13.7 1.37
Rep 4 13.7 1.37
Rep 5 13.6 1.36
Rep 6 13.7 1.37
Average (s) 1.37
Length 2 (cm) 80cm
Mass (g) 90g
Amplitude (in cm): 30cm
Time for 10 full cycles (s) Period (s)
Rep 1 18.9 1.89
Rep 2 18.8 1.88
Rep 3 18.8 1.88
Rep 4 18.9 1.89
Rep 5 18.9 1.89
Rep 6 18.9 1.89
Average (s) 1.89

Length 3 (cm) 120cm
Mass (g) 90g
Amplitude (cm) 30cm
Time for 10 full cycles (s) Period (s)
Rep 1 22.8 2.28
Rep 2 22.7 2.27
Rep 3 22.7 2.27
Rep 4 22.7 2.27
Rep 5 22.7 2.27
Rep 6 22.8 2.28
Average (s) 2.27
Overall Results
Length 1 (cm) 40cm Period (s) 1.37
Length 2 (cm) 80cm Period (s) 1.89
Length 3 (cm) 120cm Period (s) 2.27

Graphical Analysis
Graph of length, L (x-axis, independent variable) against period, T (y-axis, dependent variable)

Slope of the curve: Slope=Δy/Δx= (2.23-0.95)/(120-0)
slope=0.012 s/cm or 1.2s/m
To form an equation, we need to know the y-intercept. At y-intercept, x=0. With the gradient already known, we can get the y-intercept as follows.
0.012=(2.23-y)/(120-0)
Therefore, y-intercept is at y=0.95
Hence, the equation of the line will be: y=0.012x+0.95
The slope implies that period (T) is directly proportional to the pendulum length (L).
This finding proves my initial prediction.
Therefore, as the pendulum length increases, the period in SHM also increases.

Part D: Data Evaluation
The general formula use to find the period, T in SHM is;
T=2π√((l)/g), where l is pendulum length, g is acceleration due to gravity (9.8m/s2), and T is the period for a single oscillation.
From the experiment results in part C, we used the lengths 40cm, 80cm, and 120cm to obtain the periods 1.37s, 1.89s, and 2.27s respectively
When we use the above formula and the same pendulum lengths in the experiment, we get the following results.
For 40cm=0.40m, T=2*3.142√((0.4)/9.8)= 1.2696 seconds
For 80cm=0.80m, T=2*3.142√((0.8)/9.8)= 1.7954 seconds
For 120cm=1.20m, T=2*3.142√((1.2)/9.8)= 2.1989 seconds
The computed and the experiment values slightly differ by few microseconds. These may be due to the factors such as air turbulence, human error that may result to slightly early or late timings, and errors due to accuracy allowance in scale balance or a measuring rule.
If we were at the moon, the period of the longest pendulum (120cm) would be:
T=2*3.142√((1.2)/1.6)= 5.442 seconds
In order to swing a pendulum that weighs 1.5kg in a period of exactly 2 seconds, the pendulum must have a length of:
l=gT^2/4π2
l=9.8*4/4π^2=0.9921m
Summary: From the results of the experiment, the pendulum length, mass, and amplitude are directly proportional to the time taken for a single oscillation. If the pendulum equation is used to compute period at different lengths, the values slightly differ from those obtained from the experiment due to air turbulence, approximation errors as well as equipment accuracy allowances.

Reference
Halliday, D., Walker, J. and Resnick, R., 2010. Fundamentals of Physics, Chapters 33-37. John Wiley & Sons.

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