Question

A uniform sphere of mass $m$ and radius $r$ rolls without slipping down a inclined plane, inclined at an angle $45^{\circ}$ to the horizontal. Find the magnitude of frictional coefficient at which slipping is absent :

• A. $\frac{1}{3}$
• B. $\frac{2}{7}$
• C. $\frac{1}{5}$
• D. $\frac{1}{7}$

Answer 1

Answer: (B)

Let friction is acting in backward direction.

$ M g \sin \theta-f =M a \,\,\,\, ...(1)$
$f R =I \alpha $
$ \Rightarrow f R =\frac{2}{5} M R^{2} \times \frac{a}{R} $
$ \Rightarrow f =\frac{2}{2} M a \,\,\,\,...(2)$
$M g \sin \theta=M a+\frac{2}{5} M a=\frac{7}{5} M a$
$\Rightarrow a=\frac{5}{7} g \sin \theta$
$\Rightarrow f=\frac{2}{5} M \times \frac{5}{7} g \sin \theta$
$\Rightarrow f=\frac{2}{7} M g \sin \theta$
For pure rolling,
$f \leq \mu M g \cos \theta$
$\Rightarrow \frac{2}{7} M g \sin \theta \leq \mu M g \cos \theta$
$\Rightarrow \mu \geq \frac{2}{7}$