Vertical Oscillations of a Spring Mass System

Let’s say we have rigid support and a string hanging from the support. Let’s say we then attach a mass, there will be a small elongation, e, in the spring.

At this point the spring-mass system is in equilibrium, therefore, all the forces acting will cancel out each other, i.e the net force = 0.

F_net = 0

The two forces are the restoring force acting upwards and mg acting downwards and their net force is zero.

∴ F + mg = 0

F = -mg

We know F = –ke

–ke + mg = 0

∴ –ke = -mg

∴ ke = mg ………. *

From the equilibrium position, if a downward force is applied, the attached mass will be displaced vertically through a distance of x and the system will oscillate with an acceleration, a.

Restoring force, F’ = -k(e + x) ……..(1)

The net force towards centre = F_net

F_net = F’ + mg

F_net = -k(e + x) + mg …..(2)