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 the given situation sphere is moving upwards, it means it's slyding downwards and hence a frication force will act at upward direction,
so, option A and B are wrong.

drawing free body diagram,
$mgsin\theta -f=m{a}_{cm}$ ....(1)
$fR=I\alpha$ =$\frac{2}{5}M{R}^2\times \frac{a_{cm}}{R}m{a}_{cm}=\frac{5}{2}f$ ...(2)

on putting values of (2) into (1)
$mgsin\theta -f=\frac{5}{2}fmgsin\theta =\frac{7}{2}ff=\frac{2}{7}mgsin\theta$

Hence, option D is correct.