A uniform sphere of mass is given some angular velocity about a horizontal axis through its center and gently placed on a plank of mass m. The coefficient of friction between the two is my. The plank rests on smooth horizontal surface. The initial acceleration of the sphere relative to plank will be?
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Considering sphere of mass M is given an angular velocity ω about its horizontal axis passing through center.
Torque acting on the sphere, τ= Iα, where I is moment of inertia of sphere
also => F.R= Iα, Where F= frictional force
=> F.R=I.α
=> F.R= (5/2R) mR²α
=> α= (MmgR)/ (2/5mR²)
= (55/2R) Mg
Initial acceleration of the sphere about center of mass is
a₁ = αR= (5/2R) Mg×R
= (5/2) Mg
Acceleration for plank is
a₂= Mmg/(m+m)= Mmg/2m
= Mg/2
Therefore relative acceleration of te center of mass of the sphere & plank be
a(relative)= a₁-a₂= {(5/2)Mg - (Mg)/2}
= 4Mg/2
= 2Mg
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