Question

Show that the motion of simple pendulum is hormonic expression for it​

Answer 1

Simple harmonic motion is a special type of periodic motion ,in which a particle moves to and fro repeatedly about a mean position under a restoring force which is always directed towards the mean position.

Also the magnitude of the force at any instant is directly proportional to the displacement of the particle from the mean position at that instant.

As we can see from the figure that in the motion of a simple pendulum, it obeys both the above conditions.

Now from the above figure, weight mg of bob acts downwards and is resolved into rectangular components

Force mgsinӨ tends to bring the bob back to the mean position.

Hence restoring force F = -mgsinӨ

Ө is small hence we may say that sin Ө = Ө (Small angle approximation)

Now Ө = length of arc(x)/l where l is the length of the string

Hence F = -mgx/l.

Hence we may note here

F ∝ displacement (x)

and F is directed towards the mean position . if the bob is left there it will start executing SHM about the mean position.