A particle of mass m is attached by a light inextensible string of length l to a fixed point O. It oscillates in a vertical plane under the force of gravity through a small angle .
We have to fined the period of oscillation .
Let O be the point of suspension,
OA the vertical .
At any time t, let P be the position of the particle,
Angle AOP being theta redians and
arc AP being s.
The force that act on the particle are its weight mg vertically downwards and T the tension of the string along PO.
Resolving along the tangent PQ to the circle
m(d/dt)(ds/dt)=-mg sin theta ------ (1)
The amplitude of the oscillation is small and
therefore sin theta=theta
since ; s=theta l ,
equation (1) reduces to
(d/dt)(ds/dt)=gs/l -------------- (2)
Thus the motion is simple harmonic and the period of a Complete Oscillation is
2 pai multiplied by under root of (l/g)