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.