Question

A sphere has to purely roll upwards. At an instant when the velocity of sphere is $v$, frictional force acting on it is

• A. Downwards and $\mu mg\cos \theta$
• B. Downwards and $\frac{2mg\sin \theta }{7}$
• C. Upwards and $\mu mg\cos \theta$
• D. Upwards and $\frac{2mg\sin \theta }{7}$

Answer 1

The correct option is D Upwards and $\frac{2mg\sin \theta }{7}$

In case of rolling in the inclined plane, friction is static and acts in the upward direction and is given by
$\begin{array}{}f=\frac{mg\sin \theta }{1+\frac{R^2}{k^2}}& \text{(1)}\end{array}$
For sphere, $\begin{array}{}{k}^2=\frac{2}{5}{R}^2& \text{(2)}\end{array}$
From Eqs. (i) and (ii), $f=\frac{2}{7}mg\sin \theta$ (upwards)