The Simple Pendulum

Recall from lecture that a pendulum will execute simple harmonic motion for small amplitude vibrations. Recall from lecture that a pendulum will execute simple harmonic motion for small amplitude vibrations. Period (T) - time to make one oscillation Period (T) - time to make one oscillation Frequency (f) - number of oscillations per unit time Frequency (f) - number of oscillations per unit time

Frequency Period

In symbolic form or

The period is independent of the mass of the pendulum. The period is independent of the mass of the pendulum. The period depends on the length of pendulum. The period depends on the length of pendulum. It also depends on the amplitude (angle of swing). It also depends on the amplitude (angle of swing).

If the displacement angle is small (less than 10 0 ), If the displacement angle is small (less than 10 0 ), then the period of the pendulum depends primarily on the length ( l ) and the acceleration due to gravity (g) as follows. then the period of the pendulum depends primarily on the length ( l ) and the acceleration due to gravity (g) as follows.

It must be emphasized again that this equation is good for small angles of vibration but not for large.

Squaring both sides of the equation yields Squaring both sides of the equation yields Let’s rewrite this equation to get Let’s rewrite this equation to get

T 2 is y 4  2 /g is m l is x l is x and b will equal zero and b will equal zero This is of the form (from last week’s lab) This is of the form (from last week’s lab)

Therefore by plotting T 2 versus l and using the slope of this curve one can determine the acceleration due to gravity g. The slope is Therefore by plotting T 2 versus l and using the slope of this curve one can determine the acceleration due to gravity g. The slope is

Multiply both sides of the equation by g and get This reduces to Now divide both sides by the slope to get which reduces to

Purpose of Today’s Experiment You will determine the local value of the acceleration due to gravity by studying the motion of a simple pendulum. Note: Pendulums are used in a variety of applications from timing devices like clocks and metronomes to oil prospecting devices.