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?
The correct option is Cμ≥27√3 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] ⇒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.