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

(i) Let at some instant extension of spring is x, tension in the spring T=kx

Restoring force , F=T=-kx `" "` (Hook's law)
where , k=spring constant
Time period , `T_(0)=2pisqrt((m)/(k))`
(ii) Let block move down by x, pulley will move by x/2.

`2T=(kx)/(2)implies T=(kx)/(4)`
Restoring force, `F=-(k)/(4)x`
Comparing it with equation F=-kx
`therefore T=2pisqrt((m)/(k))impliesT_(0)=2pisqrt((m)/(k//4))=4pisqrt((m)/(k))`
(iii) If block moves by x, pulley also moves x and hence spring is stretched by 2x.

Tension produced in spring , T=4kx
Restoring force, `F=-4kx`
Comparing it with equation F=-k'x
we get k'=+4k
`therefore T=2pi((m)/(k))`
`implies T_(0)=2pisqrt((m)/(4k))=pisqrt((m)/(k))`