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# Pendulum: A Harmonic Oscillator

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Author: Janice VanCleave

A pendulum is a weight hung so that it swings about a fixed pivot (a point about which something rotates). Because of the regularity of a pendulum's swing, it is a common timekeeping mechanism.

In this project, you will determine a simple pendulum's period (time of a back-and-forth swing). You will investigate the effects of the weight and the length of the pendulum on its period. You will also use a simple pendulum to determine the acceleration of gravity and discover why the angle of displacement of a pendulum has to be small to produce simple harmonic motion.

### Getting Started

Purpose To determine the period of a simple pendulum.

### Materials

• 24-inch (60-cm) string
• metal washer
• transparent tape
• protractor
• stopwatch

### Procedure

1. Tie the string to the washer.
2. Tape the protractor to the edge of a table as shown in Figure 11.1.
3. Tape the string to the table so that the string falls across the center of the protractor.
4. Allow the pendulum to hang undisturbed. This vertical position is its resting position, labeled B in the figure. Then pull the washer to one side so that the string is at a 15° angle from vertical (position A). Release the washer and simultaneously start the stopwatch.
5. Record the time required for the washer to complete ten oscillations. Note that an oscillation is a back-and-forth swing from the displaced position A to C and back to A, indicated by arrows from A to C and from C to A.
6. Determine the period by dividing the time by 10. Record this in a Pendulum Data table like Table 11.1.
7. Repeat steps 4 through 6 four times. Average the results.

### Results

You will have determined the period for the pendulum used.

### Why?

A pendulum is a weight hung so that it swings about a fixed pivot (a point about which something rotates). A simple pendulum is a mass called a bob supported by a material, such as a string or wire of negligible mass hanging from a pivot. When a pendulum hangs vertical so that its center of gravity is below the pivot, it experiences zero net force (the sum of all forces simultaneously acting on an object) and is said to be at its resting point or in a stable equilibrium (position B in Figure 11.1). When the bob is pulled to one side, it is displaced an angular distance that depends on the amplitude of the pendulum. Displacement is the specific distance an object is moved in a specific direction. Amplitude is the farthest displacement of an object from equilibrium—the resting position for the pendulum. In this experiment the displacement angle (the angle the pendulum has moved from its resting position) is equal to 15° (angle A in Figure 11.1). The weight of the pendulum produces a restoring torque (the turning effect that reduces a pendulum's displacement angle) that moves the pendulum back toward its resting position. In Figure 11.1, f1 and f2 represent the restoring forces due to the pendulum's weight. When the pendulum is released, it exhibits periodic motion. For a pendulum, this means that it swings back and forth from position A to C and back to A about its resting position in a repetitive motion called an oscillation. The pendulum's period, T, is the time required to complete one oscillation.

At an angle of 15° or less, the periodic motion of the pendulum is considered to be simple harmonic motion (SHM), a condition in which the restoring torque is proportional to the displacement angle. This means that the restoring torque increases as the displacement angle increases. With simple harmonic motion, the period of a pendulum does not depend on amplitude.

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