A system shown in fig. (5) consists of mass less pulley, a spring of force constant k and a block of mass m. If block is just slightly displaced vertically down from its equilibrium position and released. The time period of vertical oscillation is

If m moves downward through a distance y from its equilibrium position, then pulley will also move through a distance. So, the stretching of string willbe2y. Tension T=F=k(2y) Also mg=2T ∴  Restoring force, F′=2T=4ky  Now ma=−4ky or a=−4ky/m d2ydt2=−4kmy  or  d2ydt2+4kmy=0  or d2ydt2+ω2y=0  where  ω=4km This represents S.H.M. The time period is given by T=2πω=2πm4k