A block of mass m hangs from a vertical spring of spring constant k. If it is displaced from its equilibrium position, find the time period of oscillations.

Explanation

Supose the length of the spring is stretche by a length `/_\l`. The tension in the spring is `k/_\l` and this is the force by the spring on the block. The other force on the block is mg due to gravity. For equilibrium, `mg=k/_\lor /_\l=mg/k`. Take this position of the block as x=0. If block is further displaced by x the resultant force is `k(mg/k+x)-mg=kx`.

Thus the resultnt force is proportioN/Al to the displacement. The motion is simple harmonic with a the period `T=2pisqrt(m/k)`
We see that in vertical osciltions, gravity has no effect on time period. The only effect it has is to shift the equilibrium position by a distance mg/k s if the N/Atural length is increased (or decreased if the lower end of the spring is fixed) by mg/k.