The Simple Pendulum
The simple pendulum executes approximate simple harmonic motion. A pendulum is illustrated below. At the point P all the forces acting on the pendulum are shown. The path of the pendulum is an arc of a circle since the string is inextensible.
Resolving at P for the pendulum:
Applyingalong the tangent of the arc PO: (2)
The minus sign indicates the acceleration is in the direction of decreasingEquation (1) we can ignore since the motion is along the arc of the circle and this equation is derived by resolving perpendicular to the motion.
We consider equation (2). Ifis small then
The length of a circular arc isand the acceleration of the pendulum above is(3)
If we make the substitutionthen the above equation (3) becomeswhich has the form of the simple harmonic motion equation.