Show that motion of a loaded spring is simple harmonic motion and calculate its time period of oscillation.Show that motion of a loaded spring is simple harmonic motion and calculate its time period of oscillation.

Let a point mass m be suspended from a massless spring suspended from a rigid support O.

                    
Let due to load m the spring extend by length l to acquire the equilibrium.
The restoring force set up in the spring is,
                           
where,
k is the spring constant of the spring.
Negative sign is because restoring force is in the upward direction opposite to the direction of extension which is in downward direction.
As the mass is in equilibrium, therefore

Let the mass be now pulled further by a distance y.
Now the restoring force set up in the spring is,
                   
The net force on the mass in this position is,

The acceleration produced in the mass is,

Thus, the acceleration is directly proportional to displacement and directed towards mean position.
Hence, the motion is simple harmonic motion.
The time period is given by,