Question

A spool of mass $M$ and radius $2R$ lies on an inclined plane as shown in figure. A light thread is wound around the connecting tube of the spool and its free end carries a weight of mass $m$. The value of $m$ so that system is in equilibrium is

• A. $2M\sin \alpha$
• B. $M\sin \alpha$
• C. $2M\tan \alpha$
• D. $M\tan \alpha$

Answer 1

The correct option is A $2M\sin \alpha$

frictional force will act downwards so as to balance the torque of tension force.

For rotational equilibrium, ${\tau }_o=0⟹f\left(2R\right)=TR$

For translational equilibrium: $Mgsin\alpha +f=T$

Thus $Mgsin\alpha +\frac{T}{2}=T⟹Mgsin\alpha =\frac{T}{2}$

Also, $T=mg$

$Mgsin\alpha =\frac{mg}{2}$

$⟹m=2Msin\alpha$