Suppose we have a block of unknown mass and a spring of unknown force constant. Show how we can predict the period of oscillation of this block – spring system simply by measuring the extension of the spring produced by attaching the block to it.

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.