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.

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Solution

The correct option is A

For the cyclinder, $\frac{F}{2} - f = m a_{1}$ ...(i)

Taking moment of forces about C fR= $I α$

$\Rightarrow f = \frac{m R α}{2} (∵ I = \frac{m R^{2}}{2})$ .......(ii)

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

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

$a_{1} - R α = a_{2}$ .....(iv)

from (i)and(ii),(iii)and(iv)F= $(m a_{1} + 2 m a_{2}), a n d \frac{F}{2} + \frac{m R α}{2} = 2 m a_{2}$

substititing value of $α$ and solving for $a_{1} a n d a_{2}$ , we get acceleration of plank, $a_{2} = \frac{2 F}{7 m}, a n d a c c e l e r a t i o n$

of cyclinder, $a_{1} = \frac{3 F}{7 m}$

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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.