A is a uniform solid cylinder, B is a uniform thin hollow cylinder. Both have same mass and same radius. C is a uniform plank. A, B and C each have same mass equal to m. Friction is sufficient so that there is no slipping anywhere. The time period of small oscillations of the above system is $$T=2$$$$π$$$$15mnK$$ , when the plank is disturbed slightly. Here n is an integer. Find n.

Question

A is a uniform solid cylinder, B is a uniform thin hollow cylinder. Both have same  mass and same radius. C is a uniform plank. A, B  and C each have same mass  equal to m. Friction is sufficient so that there is no slipping anywhere. The time  period of small oscillations of the above system is $$T=2$$$$π$$$$15mnK$$ , when the plank is  disturbed slightly. Here n is an integer. Find n.

Infinity Learn · Accepted Answer

Angular velocity, ω , of each cylinder will be same at a given instantLet x be the displacement of the spring from its natural, unstretched position, at a  given instantEnergy at this instant:$$E=12mR22+mR2$$ω$$2+12mR2+mR2$$ω$$2+12m(2R$$ω$$)2+12kx2E=12mR22+mR2$$ω$$2+12mR2+mR2$$ω$$2+12m(2R$$ω$$)2+12kx2$$⇒$$E=3mv24+mv2+2mv2+12kx2$$[∵$$v=R$$ω,$$v=$$ velocity of COM of cylinders ]⇒$$E=15mv24+12kx2Energy$$ of system remains constant in the absence of dissipative forces (static$$  friction on cylinders does not do $$work).  ⇒$$dEdt=0dEdt=154mva+2kvx=0$$⇒$$a=$$−$$8k15mx$$ a αx⇒$$SHMT=2$$$$π$$xa∣$$=2$$$$π$$$$15m8k$$

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