Question

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.

Answer 1

(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)`