Questions
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= , when the plank is disturbed slightly. Here n is an integer. Find n.
detailed solution
Correct option is A
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+mR2ω2+12mR2+mR2ω2+12m(2Rω)2+12kx2E=12mR22+mR2ω2+12mR2+mR2ω2+12m(2Rω)2+12kx2⇒E=3mv24+mv2+2mv2+12kx2[∵v=Rω,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=2πxa∣=2π15m8kSimilar Questions
An ideal spring with spring-constant K is hung from the ceiling and a block of mass M is attached to its lower end. The mass is released with the spring initially unstretched. Then the maximum extension in the spring is
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