  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  B
μ273  C
μ<237  D
μ<273  Solution

The correct option is C μ≥27√3Le 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] ⇒mgsinθ−f=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), mgsinθ−25ma=ma ⇒gsinθ=(a+25a) ⇒gsinθ=75a ⇒a=57gsinθ and from (3), we can say that f=27mgsinθ .....(4) For pure rolling, μN≥f From (2) and (4)  μmgcosθ≥27mgsinθ ⇒μ≥27tan θ ⇒μ≥27tan 30∘ ⇒μ≥27√3 Hence, option (b) is the correct answer. Physics

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