# Flinn PSworks™ Pendulum

## Student Laboratory Kit

### Materials Included In Kit

Metric machine screw (for PSworks™ Photogate mounting)
Plumb bob, 2 sizes, small and large
PSworks Pendulum Support Protractor
Washer, ⅛" thick

Meter stick
PSworks™ Photogate Timer
PSworks Support Stand
Scissors
Stopwatch or watch with second hand (optional)
String, thin

### Safety Precautions

The plumb bobs contain lead. Lead is extremely toxic by inhalation (dust from) and ingestion. Make sure students wash their hands after performing this experiment.

### Teacher Tips

• Enough materials are provided in this kit for one student group. 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 or hit the photogate 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.
• The additional background information and/or extension questions can be given to students based upon their instruction level and your goals for the class.
• Pendulums are excellent simple devices to use to study kinetic and potential energy. (PE = mgh, KE = ½ mv2.)
• Before students turn on the Flinn Photogate Timer, they should move the plumb bob away from the photogate. If the plumb bob remains between the gate when the photogate is turned on, it will not be possible to switch modes because the timer will begin timing. If this occurs, students should hit the Reset button and clear the data from memory.

### Further Extensions

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)?

### Science & Engineering Practices

Developing and using models
Planning and carrying out investigations
Analyzing and interpreting data
Using mathematics and computational thinking

### Disciplinary Core Ideas

MS-PS2.A: Forces and Motion
MS-PS3.A: Definitions of Energy
HS-PS2.A: Forces and Motion

### Crosscutting Concepts

Patterns
Scale, proportion, and quantity
Systems and system models
Stability and change

### Sample Data

{13311_Data_Table_1}

1. Calculate the average (mean) period times for each test in the Data Table. Record the calculations in the Data Table.

See Results Table.
Sample calculation:
Average (Mean): (1.2557 s + 1.2560 s + 1.2563 s + 1.2560 s)/4 = 1.2560 s

2. Compare the swing periods of pendulums with different starting angles. How do the different release angles affect the swing period?

The swing periods are nearly identical for pendulums of the same length, but different release angles. The period of the pendulum’s oscillation is independent of the release angle.

3. Compare the swing periods of pendulums with different masses, but the same length and starting angle. How do the different masses affect the swing period?

The swing period is not affected by the mass of the plumb bob.

4. How does the length of the pendulum affect the swing period?

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

5. Based upon the data from this experiment:

(T) F The period of a pendulum is not affected by the mass of the plumb bob on the end of the pendulum.
T (F) The period of a pendulum is affected by how high the pendulum is raised before it is released.
T (F) The period of a pendulum increases as the pendulum length decreases.
T (F) A grandfather clock will “tick-tock” faster when the pendulum is released with a large swing arc compared to a small swing arc.

Extension

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)?

### 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 4 for a diagram of the forces acting on the plumb bob.)

{13311_Discussion_Figure_4}
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 4). 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.
{13311_Discussion_Equation_1}

T = period of oscillation
L = length of pendulum
g = 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 (Equation 2).
{13311_Discussion_Equation_2}
ϴo = 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 Pub.: New York, 1990; pp. 382–385.

# Flinn PSworks™ Pendulum

### Introduction

A swinging pendulum 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 this equilibrium 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 (a back and forth motion in which the plumb bob returns to the original release point is one complete oscillation) will take the same amount of time. The time it takes for one complete oscillation is known as the period of the oscillation.

What affects the period of a pendulum? Does the release height affect the time it takes for a pendulum to complete one oscillation? What about the weight of the plumb bob? Will a heavier plumb bob swing slower or faster than a lighter plumb bob, or will they swing with the same period? Follow the experiment in this laboratory activity to determine what affects the period of a pendulum’s swing.

### Materials

Meter stick
Metric machine screw (for PSworks™ Photogate mounting)
Plumb bob, 2 sizes, small and large
PSworks Pendulum Support Protractor
PSworks Photogate Timer
PSworks Support Stand
Scissors
Stopwatch or watch with secondhand (optional)
String, thin, 75 cm
Washer, ⅛" thick

### Procedure

Assembly

1. Attach the Pendulum Support Protractor near the middle of the aluminum rod of the PSworks™ Support Stand using the knob with threaded stud as shown in Figure 1.
{13311_Procedure_Figure_1}
2. Use meter stick to measure and scissors to cut string to approximately 75 cm.
3. Tie one end of the string to one of the plumb bobs (large or small fishing sinker).
4. Slide the other end of the string into the slotted rod in the Pendulum Support Protractor and wrap the excess string around the rod to secure the pendulum as shown in Figure 2.
{13311_Procedure_Figure_2}
5. Align the 0° line of the Pendulum Support Protractor with the hanging string of the pendulum.
6. Attach the photogate to the aluminum rod using the metric machine screw and washer. Thread the screw through a hole in the aluminum rod, place the washer on the screw between the support stand and the photogate, and turn the screw into the photogate (make sure the phone cord is plugged into the output jack of the photogate and the timer). Turn the “open end” of the photogate up and screw in as tightly as possible using a screwdriver if necessary (see Figure 3).
{13311_Procedure_Figure_3}
Activity
1. Adjust the length of the pendulum so that the middle of the plumb bob crosses the path between the two slits in the photogate (the photogate timer should be turned off at this time). The pendulum string should hang as close to the edge of the slotted rod as possible. Wrap the excess string around the slotted rod in the Pendulum Support Protractor (see Figure 2).
2. Use a meter stick to measure the length of the pendulum from the point just below the slotted rod in the Pendulum Support Protractor to the middle of the plumb bob. Measure to the nearest 0.1 cm and record the measurement in meters in the Pendulum Worksheet.
3. Move the plumb bob away from the photogate and then turn on the Flinn Photogate Timer.
4. Continue to hold the plumb bob away from the photogate and press the Mode button on the Flinn Photogate Timer until Pendulum Mode is displayed.
5. Move the pendulum so that string lines up with the 5° mark on the Pendulum Support Protractor.
6. Line up the plumb bob with the photogate so that it will swing freely through the photogate without hitting it.
7. Once the plumb bob is properly aligned, carefully release the pendulum.
8. Watch the motion of the pendulum (e.g., its maximum angle, its speed during its motion) as well as the time measurements recorded by the photogate timer.
9. Stop the pendulum after 10 measurements register on the timer.
10. Hold the pendulum away from the photogate and press the Memory button on the timer. Pressing the Memory button continuously will scroll through the data just measured.
11. Record the time measurements from Memory Slots 2–5 in the Pendulum Worksheet.
12. After recording the time measurements, simultaneously press the Memory and Reset buttons to delete the data.
13. Repeat steps 3–12, but start the pendulum at a 15° angle. Record the time measurements in the Pendulum Worksheet.
14. Unscrew the photogate (or the Pendulum) from the PSWORKS Support Stand, and then reattach it three or four hole positions lower (or higher), according to Assembly step 6.
15. Unwind the string and lower the plumb bob so that the middle of the plumb bob crosses the path between the slots in the photogate. Wrap the excess string around the slotted rod in the Pendulum Support Protractor (see Figure 2).
16. Repeat steps 2–13. Record the time measurements in the Pendulum Worksheet.
17. Repeat steps 1–12 using the other (small or large) plumb bob. Make sure the length of the pendulum is nearly the same as the length used in one of the other trials. Record all the time measurements in the Pendulum Worksheet.
18. Return the materials to your instructor for future use.

### Student Worksheet PDF

13311_Student1.pdf

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