Experiment 8

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

The pendulum is a familiar oscillator in nature. A simple pendulum consists of a mass (the bob) suspended from a string 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 to the side (with the string taut) and then released, it will exhibit oscillatory or harmonic motion. The path of the motion always follows a circular arc with a 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 (θ in Fig. 8-1). 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 approximates 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 8-1. 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 string. 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. Exactly as 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, back toward equilibrium, so the pendulum slows down. The net force increases as the angle increases, and at some point the velocity of the pendulum reaches zero. The net force is pulling it back toward equilibrium, and so it moves back the way it came, 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 behavior of the pendulum can be found by solving Newton’s second law of motion, which predicts that the Period of the pendulum is given by:

Notice that the mass of the pendulum bob does not enter into this equation, which means that the Period does not depen on its mass. The only physical characteristic of the pendulum that the Period depends on is its length L. The reate at which the pendulum swings is the frequency:

In this experiment, you will study the constancy of the period for a fixed length pendulum, and study the how the period depends on the length. As a result, you can also measure the acceleration due to gravity g.