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Question

# A massless spool of inner radius r, outer radius R is placed against vertical wall and tilted split floor as shown. A light inextensible thread is tightly wound around the spool through which a mass m is hanging. There exists no friction at point A, while the coefficient of friction between spool and point B is μ. The angle between two surfaces is θ.

A
the magnitude of force on the spool at B in order to maintain equilibrium is mg(rR)2+(1rR)21tan2θ
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B
the magnitude of force on the spool at B in order to maintain equilibrium is mg(1rR)1tanθ
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C
the maintain value of μ for the system to remain in equilibrium is cotθ(R/r)1
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D
the maintain value of μ for the system to remain in equilibrium is tanθ(R/r)1
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Solution

## The correct options are A the maintain value of μ for the system to remain in equilibrium is tanθ(R/r)−1 B the magnitude of force on the spool at B in order to maintain equilibrium is mg√(rR)2+(1−rR)21tan2θmgr=fR ....(i)N1sinθ+f=mg ....(ii)N1cosθ=N2 ....(iii)From (i), f=mgrRFrom (ii) and (iii),N2=mg−ftanθ=mgtanθ[1−rR]Net force at B: FB=√f2+N22FB=mg√(rR)2+(1−rR)21tan2θFor minimum value of μ:f≤μN2⇒mgrR≤μmgtanθ[1−rR]⇒μ≥tanθR/r−1

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