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Question

A cylinder of mass M, radius R is resting on a horizontal platform (which is parallel to xyplane) with its axis fixed along the yaxis and only free to rotate about its axis. The platform is given a motion in xdirection given by x=Acos(ωt). There is no slipping between the cylinder and platform. The maximum torque acting on the cylinder during its motion is

A
12MRAω2
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B
MRAω2
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C
2MRAω2
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D
MRAω2×cosωt
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Solution

The correct option is A 12MRAω2
The axis of solid cylinder is fixed hence it will only rotate about its axis of rotation (y-axis).


Motion of platform is given by,
x=Acos(ωt)
Acceleration of the platform is given by
a=d2xdt2=Aω2cos(ωt)
Above equation represents the SHM for the platform,
The maximum acceleration achieved by the platform.
|amax|=Aω2
For no slipping between the platform and cylinder,
angular acceleration of cylinder, α=aR
where a=aplatform
For maximum magnitude of torque acting on the cylinder, magniude of α and a will be maximum.
i.e |τmax||amax|
τmax=Iαmax
Here, I MOI of the solid cylinder about axis passing through its centre i.e yaxis.
τmax=(MR22)(amaxR)
τmax=MR22×Aω2R
τmax=12MRAω2

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