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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.

120066_c42ef264098e49578ac997b5db929736.png

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+(1rR)21tan2θ
mgr=fR ....(i)
N1sinθ+f=mg ....(ii)
N1cosθ=N2 ....(iii)
From (i), f=mgrR
From (ii) and (iii),
N2=mgftanθ=mgtanθ[1rR]

Net force at B: FB=f2+N22

FB=mg(rR)2+(1rR)21tan2θ

For minimum value of μ:fμN2

mgrRμmgtanθ[1rR]μtanθR/r1

132729_120066_ans_3f792482449b44e89352daa41094f3e0.png

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