Question

A solid cylinder of mass 'm' and radius 'R' rests on a plank of mass '2m' lying on a smooth horizontal surface. String connecting the cylinder to the plank is passing over a massless pulley mounted on a movable light block B, and the friction between the cylinder and the plank is sufficient to prevent slipping. If the block B is pulled with a constant force 'F', find the acceleration of the cylinder and that of the plank.

• A. $\frac{3F}{7m},\frac{2F}{7m}$
• B. $\frac{2F}{7m},\frac{3F}{7m}$
• C. $\frac{4F}{7m},\frac{3F}{7m}$
• D. $\frac{3F}{7m},\frac{4F}{7m}$

Answer 1

The correct option is A

$\frac{3F}{7m},\frac{2F}{7m}$

For the cyclinder, $\frac{F}{2}-f=m{a}_1$ ...(i)

Taking moment of forces about C fR=$I\alpha$

$\Rightarrow f=\frac{mR\alpha }{2}(\because I=\frac{m{R}^2}{2})$ .......(ii)

For the plank, $\frac{F}{2}+f=\left(2m\right)a_2$ ......(iii)

And at the point of contact P, the acceleration of the two bodies must be same

$a_1-R\alpha =a_2$ .....(iv)

from (i)and(ii),(iii)and(iv)F=$(m{a}_1+2m{a}_2),and\frac{F}{2}+\frac{mR\alpha }{2}=2m{a}_2$

substititing value of $\alpha$ and solving for $a_{1}and{a}_2$, we get acceleration of plank, $a_2=\frac{2F}{7m},andacceleration$

of cyclinder, $a_1=\frac{3F}{7m}$