A solid sphere of mass M and radius R is kept on a rough surface. The velocities of air (density ρ ) around the sphere are as shown in figure. Assuming R to be small and M = g 4 π ρ R 2 kg , what is

Question

A solid sphere of mass M and radius R is kept on a rough surface. The velocities of air (density ρ ) around the sphere are as shown in figure. Assuming R to be small and M = g 4 π ρ R 2 ​ kg , what is the minimum value of coefficient of friction so that the sphere starts pure rolling? (Assume force due to pressure difference is acting on centre of mass of the sphere) :

TestHub · Accepted Answer

Force due to pressure difference: F = 2 1 ​ ρ π R 2 ( v 2 2 ​ − v 1 2 ​ ) = 2 1 ​ ρ π R 2 ( 14 − 7 ) = 2 7 ​ ρ π R 2 F − f = M a ; torque about centre: f R = 5 2 ​ M R 2 ⋅ R a ​ ⇒ f = 5 2 ​ M a = 7 2 ​ F = ρ π R 2 Just rolling: μ M g = ρ π R 2 μ = M g ρ π R 2 ​ = 4 1 ​ = 0.25 (using M = g 4 π ρ R 2 ​ ) - (A)

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