Materials Included In Kit

Additional Materials Required

Safety Precautions

The plumb bobs contain lead. Lead dust is extremely toxic by inhalation and ingestion. Remind students to wash their hands thoroughly with soap and water after performing this experiment. Please review current Safety Data Sheets for additional safety, handling and disposal information.

Disposal

Materials should be saved for future use.

Teacher Tips

• Enough materials are provided in this kit for 30 students working in pairs or for 15 groups of students. This laboratory activity can reasonably be completed in one 50-minute class period.
• To protect against the hazards of lead, the plumb bobs can be dipped into melted wax or coated with a clear coat of paint. The additional mass will not affect the results of the experiment.
• Advise students to release the pendulum so that it swings back and forth in one plane and does not rotate as it oscillates.
• Students should practice releasing the pendulum so that it swings smoothly. Lightly holding the plumb bob from the bottom with only a fingertip as it is pulled to the appropriate release angle and then lowering the finger to release the bob will provide a smooth release and even swing.
• If protractors are not available, simple triangles can be drawn on separate sheets of paper to create 5° and 15° angles. For the 5° angle, the sides of a right triangle should measure 2.2 cm and 25 cm. The small angle of this triangle (the angle between the long side of the triangle and the hypotenuse) will measure 5°. For the 15° angle, the sides of a right triangle should be 6.7 cm and 25 cm.
• Students can work in pairs with one student measuring the angle of the pendulum with a protractor and releasing the plumb bob, while the other student times and counts the oscillations. Also, a student can perform the experiment individually by first using a protractor to measure the appropriate angle. While holding the pendulum at this angle with one hand, the student will then need to place the protractor down in order to start the stopwatch. A clock with a second hand can also be used as the timing device.
• The additional background information and/or extension questions can be given to students based upon their instruction level and your goals for the class.
• When students use Equation 1 to solve for the acceleration of gravity constant from the slope of the graph, make sure students understand that the entire equation is squared. Therefore, the slope is equal to 4π²/g, not 2 π/g.
• Pendulums are excellent simple devices to use to study kinetic and potential energy. (PE = mgh, KE = ½mv².)

Further Extensions

Extension Post-Lab Questions

1. From Equation 1, what is the formula for the slope of the best-fit line from Post-Lab Question 5? Hint: The formula of a straight line going through the origin (0,0) is y = mx, where m is the slope. Determine the slope of the best-fit line and calculate the acceleration of gravity constant, g.
2. What is the period of oscillation for a 1-m-long pendulum, with a 100-g plumb bob, on the surface of the Earth (g = 9.81 m/s²)? What would the period of oscillation be on the moon (g = 1.62 m/s²)? 

Sample Data

Experiment 1

Experiment 2 Sample Calculations

Results Table—Experiment 1 Results Table—Experiment 2

Answers to Questions

1. Calculate the average (mean) number of oscillations in 30 seconds for each test in Experiment 1 and 2. Then, calculate the oscillation period for each test by dividing 30 seconds by the average number of complete oscillations in 30 seconds from each test. (Optional) For Experiment 2, calculate the oscillation period squared [(oscillation period)²]. Record the calculations in the results table.

   See results tables.
   
   Sample calculations:
   Average (Mean): (21 + 22 + 21)/3 = 21.3333 ——> 21
   Period: 30 s/21 = 1.429 s ——> 1.4 s
   Period squared: (1.4 s)² = 1.4 s x 1.4 s = 1.96 s² ——> 2.0 s²

2. Compare the oscillation periods in Experiment 1. How do the different release angles affect the oscillation period? How do the different masses affect the oscillation period?

   The oscillation periods are nearly identical for all the trials of Experiment 1. The period of the pendulum’s oscillation is independent of the release angle and the mass of the plumb bob.

3. Compare the oscillation periods in Experiment 2. How does the length of the pendulum affect the oscillation period?

   The oscillation period of the pendulum increased with increasing pendulum length. The increase is not proportional to the length.

4. Based upon the data from this experiment:
   {12034_Answers_Table_5}
5. (Optional) On a separate sheet of paper, make a graph with the oscillation period squared on the y-axis versus the pendulum length on the x-axis. Plot the data and draw a best-fit line through the data points starting from the origin (0,0). Is the best-fit line a straight line? If so, what does this say about the relationship between the period of the oscillation and the length of the pendulum?
   {12034_Answers_Figure_4}
   Yes, the best-fit line is straight. This means that the period of the pendulum is proportional to the square-root of the pendulum length. T ∝ √L

Extension Post-Lab

1. From Equation 1, what is the formula for the slope of the best-fit line from Post-Lab Question 5? Hint: The formula of a straight line going through the origin (0,0) is y = mx, where m is the slope. Determine the slope of the best-fit line and calculate the acceleration of gravity constant, g.

   The slope of the best-fit line is inversely proportional to the acceleration of gravity constant, g.
   From the graph, the slope of the best-fit line can be calculated as the rise over the run of the best-fit line.
   
   From the data:
   Slope of best-fit line = 2.0 s² – 0.80 s²/0.50 m – 0.20 m = 4.0 s²/m
   g can be calculated by squaring Equation 1: T² = 4π²L/g
   4π²/g = slope of best-fit line
   g = 4π²/slope of best-fit line
   g = 4π²/4.0 s²/m = 9.9 m/s²

2. What is the period of oscillation for a 1-m-long pendulum, with a 100-g plumb bob, on the surface of the Earth (g = 9.8 m/s²)? What would the period of oscillation be on the moon (g = 1.6 m/s²)?
   {12034_Extension_Equation_3}

Discussion

A simple pendulum is composed of string tied to a rigid object at one end (the anchoring point) with a freely hanging mass (m), also known as a plumb bob, tied to the other end. When the pendulum is at rest, the plumb bob will hang directly below the anchoring point, and the string will be vertical. The only external forces acting on the plumb bob are from the pull of gravity (mg) and from the tension in the string (T) holding the plumb bob up. When the pendulum is vertical, these forces are balanced. When the plumb bob of the pendulum is moved away from its equilibrium (at rest) position along the arc of the pendulum swing and then released, gravity and the tension in the string are still the only forces acting on the plumb bob. However, now these forces are no longer balanced. The unbalanced forces result in a restoring force (mg sin θ) that moves the plumb bob back toward the equilibrium position along the arc of the swing. Because of momentum, however, the plumb bob will continue to swing past the equilibrium position. Once it passes equilibrium, the plumb bob will swing up along the pendulum’s arc and a restoring force will again act on the plumb bob to slow it down until it momentarily stops, and then falls back down towards its equilibrium position and the cycle will repeat itself. The pendulum will continue to oscillate back and forth this way indefinitely if no other forces (such as friction) act on it. (See Figure 3 for a diagram of the forces acting on the plumb bob.)

For small displacements, the restoring force acting on the plumb bob is directly proportional to the displacement away from the equilibrium position. That is, the farther away from equilibrium, the larger the restoring force. As the plumb bob swings closer to equilibrium, the restoring force decreases evenly. When the restoring force is directly proportional to the displacement, the oscillations are said to exhibit simple harmonic motion. In simple harmonic motion, the pendulum will oscillate back and forth along an arc following the same path and reach the same displacement away from equilibrium each time. The time it takes for each complete oscillation will be constant. The displacement away from equilibrium is also called the amplitude of the oscillation (θ in Figure 3). The time of each complete oscillation is known as the period of the oscillation. In simple harmonic motion, the period of oscillation is given by Equation 1. where

T is the period of oscillation
L is the length of pendulum
g is the acceleration of gravity constant

This equation shows that a pendulum’s swing is independent of the mass of the plumb bob and the amplitude of the swing. It depends only on the length of the pendulum and the acceleration of gravity.

As long as the amplitude is relatively small, the oscillations will exhibit simple harmonic motion. The oscillations will exhibit a linear relationship that does not depend on the amplitude. However, as the amplitude increases, the oscillations of the pendulum will no longer appear to be simple harmonic. The oscillations will follow a non-linear relationship (see Equation 2).

θ_o is the release angle (initial amplitude)

However, the period of a pendulum released at 20 degrees from equilibrium will still vary by less than 1% from the “ideal” simple harmonic motion period.

References

Tipler, Paul A. Physics for Scientists and Engineers, 3rd ed., Vol. 1; Worth: New York, 1990; pp 382–385.

Introduction

A swinging pendulum is a very common sight—from children swinging on a swing set to the timekeeping swing of a grandfather clock. It is the simplest oscillating system. Let’s explore the properties of a pendulum’s swing.

Concepts

• Pendulums
• Gravity
• Period of oscillation
• Simple harmonic motion

Background

A simple pendulum is composed of a string tied to a rigid object at one end (the anchoring point) with a freely hanging mass, also known as a plumb bob, tied to the other end. When the pendulum is at rest, the plumb bob will hang directly below the anchoring point, and the string will be vertical. When the pendulum is displaced away from the equilibrium (at rest) position and released, the force of gravity will cause the pendulum to swing back and forth along a swing arc. The pendulum will oscillate with simple harmonic motion. That is, the pendulum will oscillate back and forth along an arc, following the same path, reach the same displacement away from the equilibrium position, and each complete oscillation will take the same amount of time. Note: One complete oscillation is a back and forth motion in which the plumb bob returns to the original release point. The time it takes for one complete oscillation is known as the period of the oscillation.

Experiment Overview

The purpose of this activity is to determine what affects the period of a pendulum’s swing. Variables tested will include the weight of the plumb bob, the release angle and the length of the pendulum.

Materials

Safety Precautions

The plumb bobs contain lead. Wash hands thoroughly with soap and water before leaving the laboratory. Please follow normal laboratory safety guidelines.

Procedure

Prelab Setup

1. Obtain a support stand and ring.
2. Cut a length of string 75 cm long.
3. Tie the small plumb bob to the end of the string. Keep excess string to a minimum or cut off the excess string from the knot.
4. Clamp the string to the side of the ring with the clothespin.
5. Position the support stand so that the plumb bob can dangle over the edge of the table and swing freely (see Figure 1).
   {12034_Procedure_Figure_1_Pendulum setup}

Experiment 1: Measure the Period of the Pendulum Swing

6. Use a meter stick and adjust the length of the pendulum so that it is 50‑cm (0.50 m) from the bottom of the plumb bob to the bottom of the support stand ring. Use your finger to hold the string in place on the ring and then wrap the excess string around the ring two or three times so the string does not slip. Then secure the string by clamping the excess string looped around the ring with a clothespin clamp (see Figure 2).
   {12034_Procedure_Figure_2_Clamp setup}
7. Pull the plumb bob along its swing arc to an angle of 5 degrees from equilibrium. Use a protractor to measure the angle of the string.
8. With a stopwatch or watch with second hand ready, release the plumb bob and begin timing. Count the number of times the plumb bob swings through one complete oscillation for a total time of 30 seconds. Note: The easiest way to count is to count how many times the plumb bob returns to the initial release point.
9. Record the number of complete oscillations under trial 1 in the data table for Experiment 1.
10. Repeat steps 7–9 two more times for trials 2 and 3.
11. Next, pull the plumb bob along its swing arc to an angle 15 degrees from equilibrium.
12. Release the plumb bob and begin timing. Count the number of times it swings through one complete oscillation for 30 seconds. Record the number of complete oscillations in the appropriate space the data table.
13. Repeat steps 11 and 12 two more times for trials 2 and 3.
14. Untie or cut the string to remove the small plumb bob and then tie the large plumb bob to the end of the string.
15. Repeat steps 6–13. Record your data in the appropriate spaces in the data table.

Experiment 2: The Period of the Pendulum Versus the Length of the Pendulum

16. With the large plumb bob tied to the string, adjust the pendulum length to 20 cm (0.20 m).
17. Pull the plumb bob along its swing arc to an angle of 5 degrees from equilibrium.
18. With a stopwatch or watch with second hand ready, release the plumb bob and begin timing. Count the number of times it swings through one complete oscillation for a total time of 30 seconds.
19. Record the number of complete oscillations in the appropriate space in the data table for Experiment 2.
20. Repeat steps 17–19 two more times for trials 2 and 3.
21. Repeat steps 16–20 using both 30-cm and 40-cm pendulum lengths. Use the data from the Experiment 1 for the 50-cm pendulum length (large plumb bob from a release angle of 5 degrees from equilibrium).
22. Return the plumb bobs and cut string to your instructor for future use.

Student Worksheet PDF

12034_Student1.pdf