First slide

Rolling motion

Question

A uniform solid sphere rolls down a vertical surface without sliding. If the vertical surface moves with an acceleration $a = \frac{g}{2}$ , the minimum coefficient of friction between the sphere and vertical surfaces so as to prevent relative sliding is

Difficult

A

3/7
B

4/7
C

4/5
D

3/5

Solution

Let sphere rolls down with acceleration a'. Writing force and torque equations for sphere
$\begin{array}{l} N = \frac{m g}{2} . . . . . (i) \\ m g - f = m a^{'} . . . . . . (i i) \\ f R = (\frac{2}{5} m R^{2}) \cdot α \\ f = \frac{2}{5} m R α . . . . . . . (i i i) \\ α R = a^{'} . . . . . . . (i v) \end{array}$
$\begin{array}{l} \Rightarrow f = \frac{2}{5} m a^{'} or a^{'} = \frac{5}{2} \frac{f}{m} . . . . . . . . (v) \\ from (ii) and (v) \\ m g - f = \frac{5}{2} f \Rightarrow m g = \frac{7}{2} f or f = \frac{2}{7} m g \\ But, f \leq N hence \frac{2}{7} m g \leq \frac{μ m g}{2} \Rightarrow μ \geq \frac{4}{7} \end{array}$

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