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PURPOSE

To investigate the time-dependent characteristics of the periodic motion of a pendulum.

EQUIPMENT  pendulum, assorted masses, mass holder, photogate, DataLogger interface, meter stick

RELEVANT EQUATIONS

	(s)
	(Hz)

DISCUSSION

The pendulum is a familiar oscillator in nature. A simple pendulum consists of a mass (the bob) suspended from a cord of negligible mass. The equilibrium position is where the pendulum is perfectly vertical under the influence of gravity. If the pendulum is displaced from its equilibrium position by pulling it aside along the circular arc described by its constant length, and then released, it will exhibit oscillatory, or harmonic motion. The path of the motion is always constrained to follow a circular arc with radius equal to the string length L.

The maximum displacement from equilibrium is called the amplitude, which in the case of a pendulum, is expressed as an angle. The time necessary for the pendulum to execute one complete oscillation, or cycle, is called the period. The term Simple Harmonic Motion refers to any harmonic oscillator whose period does not depend on the amplitude of the motion. The simple pendulum is a Simple Harmonic Oscillator only for amplitudes of 6° or less.

The oscillating motion of the pendulum can be understood by examining a force diagram, as illustrated in Figure 20-1. The component of the bob's weight that is directed along the circular arc makes up the net force, because the other component of the weight exactly balances the tension in the cord. The net force produces acceleration toward the equilibrium position.

As the pendulum moves along the arc toward the vertical equilibrium line, the net force on it gets smaller and smaller. At the exact instant that it passes through the equilibrium position, the net force on the pendulum is zero. It has, however. picked up speed as it was being accelerated, and so continues to move on past the equilibrium position. On the other side of the equilibrium line, the net force is directed in the opposite direction along the arc. The pendulum experiences a deceleration and slows down. The net force increases as the angle increases, and at some point, the instantaneous velocity of the pendulum reaches zero. The net force is pulling it back toward equilibrium, and so it moves in the opposite direction, repeating its motion over and over again. The net force in this context is called a restoring force, because it always acts in such a way as to restore the pendulum back to its equilibrium position.

The time behavior of the pendulum can be described by solving Newton's second law of motion. To summarize, this solution predicts that the period of the pendulum is given by:

(1)

Notice that the mass of the pendulum bob does not enter into this equation. This result means that the period is independent of its mass. The only physical characteristic of the pendulum that the period depends on is its length L. An alternative description of the temporal behavior of the pendulum is the frequency:

(2)

In this experiment, you will study the constancy of the period for a fixed length pendulum, and study the dependence of the period T, on the length L as it is varied. As a result, you can determine the acceleration due to gravity, g.