Question

A plank of mass $M$ is placed over smooth inclined plane and a sphere of mass $m$ is also placed over the plank. Friction is sufficient between sphere and plank. If plank and sphere are released from rest, the frictional force on sphere is :

• A. up the plane
• B. down the plane
• C. horizontal
• D. zero

Answer 1

The correct option is D zero

Let the friction force acting on sphere is up the plane. (If its down the plane then answer will come negative).
From FBD of block, force balance along the plane
$M{a}_1=Mgsin\theta +f$
or
$a_1=gsin\theta +\frac{f}{M}$ -(I)
With respect to block let the acceleration of sphere be $a_2$
Therefore angular acceleration of sphere is given by: $\alpha \times r=a_2$
Torque on sphere= $f\times r=I\times \alpha =\frac{m{r}^2}{2}\alpha =\frac{mr{a}_2}{2}$
Therefore $f=\frac{m{a}_2}{2}$
$a_2=\frac{2f}{m}$ -(II)
Force balance along the plane for sphere with respect to block:
$m{a}_2=mgsin\theta -f-m{a}_1$
From (I) & (II)
$3f=mgsin\theta -m(gsin\theta +\frac{f}{M})=-f\frac{m}{M}$
Therefore $f=0$