A block suspended from a vertical spring is in equilibrium. Show that the extension of the spring equals the length of an equivalent simple pendulum, i.e., a pendulum having frequency same as that of the block.

Let the Extension produced in the Spring be A.

Therefore, In equilibrium position,

Force due to Gravity = Spring Force.

Now, Time Period of the Spring-mass system

Now, Time period for Simple Pendulum

Now, Frequency is equal, therefore, Time period is also equal.

Equating the Time Period of both the cases,

After Cancelling the terms and applying simple mathematics,

l = A,

this means the extension in the Spring equals to the length of the time period.