Simple Pendulum

Show For pendulum length L = cm = m and acceleration of gravity g = m/s^2 the pendulum period is T = s (Enter data for two of the variables and then click on the active text for the third variable to calculate it.) This expression for period is reasonably accurate for angles of a few degrees, but the treatment of the large amplitude pendulum is much more complex.

If the rod is not of negligible mass, then it must be treated as a physical pendulum.

Pendulum Motion

The motion of a simple pendulum is like simple harmonic motion in that the equation for the angular displacement is
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which is the same form as the motion of a mass on a spring:

The anglular frequency of the motion is then given by

compared to

for a mass on a spring.

The frequency of the pendulum in Hz is given by

and the period of motion is then

.

Period of Simple Pendulum

Show For small angles , we can use the approximation Show in which case Newton's 2nd law takes the form Even in this approximate case, the solution of the equation uses calculus and differential equations. The differential equation is and for small angles the solution:

Pendulum Geometry

Pendulum Equation