In the following arrangements, bock is slightly displaced vertically down from its equilibrium position and released. Find time period of vertical oscillations. The pulley is light.

Explanation

(a)
Let at some instant extension of spring is x, tension in the string `T=kx`
Restoring force `F=T=-kx`
Time period `T_0=2pisqrt(m/k)`
(b)
Let block move down by x, pulley will move by `x//2`
`2T=(kx)/(2)impliesT=(kx)/(4)`
`F=-k/4x`
`T_0=2pisqrt((m)/(k//4))=4pisqrt(m/k)`


If block moves by x, pulley also moves x, spring is stretched by `2x`
`T=4kx`
`F=-4kx`
`T_0=2pisqrt((m)/(4k))=pisqrt(m/k)`