Simple harmonic motion (SHM) is the motion in which the restoring force (F) is proportional to displacement (x) from the mean position and opposes its increase.
So, F = kx
Here, k is the force constant.
The restoring force F acting on the body is
F = mass × acceleration
= ma
When the block of mass m is suspended with the spring, then at equilibrium position, the restoring force must be equal to the weight (mg) of the body.
So,
kx = mg
k/m = g/x
We know that that angular velocity (w) is defined as,
w = √k/m
To find out the angular velocity (w) in terms of g and x, substitute g/x for k/m in the equation w = √k/m,
w = √k/m
= √ g/x
But the time period (T) of the oscillation is defined as,
T = 2π/ w
To find out the time period, substitute √ g/x for w in the equation T = 2π/ w,
T = 2π/ w
= 2π√x/g
So, by measuring the displacement of the spring when it is attached with a block of mass m and suspended freely, you can predict the period of oscillation of this block.