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

Question

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

Infinity Learn · Accepted Answer

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 $$d2ydt2+$$ω$$2y=0$$  where  ω$$=4km$$ This represents S.H.M. The time period is given by $$T=2$$πω$$=2$$$$π$$$$m4k$$

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

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