A simple pendulum is an arrangement in which a point mass is suspended by an inextensible weightless string in a uniform gravitational field. This is an ideal system which cannot be realized in practice. However, a pendulum consisting of a small but relatively heavy bob on one end of a very light string can be considered as a simple pendulum.
In its equilibrium position, the thread will be vertical with the weight mg of the bob being balanced by the tension in the string. If the bob is pulled aside and released, it oscillates about its equilibrium position.
Let P1 represent the position of the bob when it is displaced through a small distance x. The component of mg tangential to the arc PP1 provides the restoring force.
Applying the formula s = r q for the arc PP1, we get
The motion is approximately simple harmonic. Comparing this equation with the standard equation from SHM, namely
F = -Kx, we get
The period in SHM is given by
Clocks are usually fitted with pendulums which take one second to move from one end of their swing to the other end. Since period is the time taken for a to and fro motion, the period of a second's pendulum is 2 seconds. Let L represent the length of such a pendulum. Then substituting in equation (13), we get
Squaring and rearranging, we get
