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Question

A solid sphere A of mass m rolls without slipping on an inclined plane of inclination 30. Coefficient of friction of the inclined plane is μ. Then, which of the following options is satisfied when the sphere undergoes pure rolling motion?


A
μ237
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B
μ273
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C
μ<237
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D
μ<273
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Solution

The correct option is C μ273
Le us assume the radius of the sphere is R.
F.B.D from the data given in the question is shown below.


From the diagram, we can deduce that
Fx=max [Net Force along x - direction]
mgsinθf=ma ......(1)
Fy=may [Net Force along y - direction]
N=mgcosθ .....(2)
Torque experienced by the sphere due to frictional force is given by 
fR=Iα
where I= Moment of inertia of solid sphere =25mR2 
fR=(25mR2)(aR) [α=aR] 
f=25ma ......(3)
Substituting (3) in (1),
mgsinθ25ma=ma
gsinθ=(a+25a)
gsinθ=75a
a=57gsinθ
and from (3), we can say that
f=27mgsinθ .....(4)
For pure rolling, μNf
From (2) and (4) 
μmgcosθ27mgsinθ
μ27tan θ
μ27tan 30
μ273
Hence, option (b) is the correct answer. 

Physics

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