First slide

Rolling motion

Question

An imperfectly rough sphere moves from rest down a plane inclined at an angle $α$ to the horizontal. The coefficient of friction between the inclined plane and sphere is $μ .$ Then

Moderate

A

if $μ < \frac{2}{7} \tan α,$ $μ > \frac{1}{7} \tan α,$ then pure rolling is not possible.
B

if $μ > \frac{1}{7} \tan α,$ then only pure rolling takes place.
C

if $μ > \frac{2}{7} \tan α,$ then pure rolling takes place.
D

all of the above are wrong.

Solution

$For pure rolling μ \geq tanα / (1 + \frac{{mr}^{2}}{I_{c}}) For solid sphere, I_{C} = \frac{2}{5} {mr}^{2} ∴ μ \geq tanα / (1 + \frac{5}{2}) \Rightarrow μ \geq \frac{2}{7} tanα \Rightarrow μ_{\min} = \frac{2}{7} tanα$

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