A man pushes a cylinder of mass $m_{1}$ with the help of a plank of mass $m_{2}$ as shown. There is no slipping at any contact. The horizontal component of the force applied by the man is $F$ . Find the acceleration of the plank and the centre of mass of the cylinder.

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$a_{1} = \frac{8 F}{3 m_{1} + 8 m_{2}}, a_{2} = \frac{2 F}{3 m_{1} + 8 m_{2}}$

$a_{1} = \frac{4 F}{3 m_{1} + 4 m_{2}}, a_{2} = \frac{2 F}{3 m_{1} + 2 m_{2}}$

$a_{1} = \frac{F}{m_{1} + 8 m_{2}}, a_{2} = \frac{4 F}{m_{1} + 8 m_{2}}$

$a_{1} = \frac{4 F}{3 m_{1} + 8 m_{2}}, a_{2} = \frac{8 F}{3 m_{1} + 8 m_{2}}$

answer is B.

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Detailed Solution

For the motion of the plank, we have

$F - f_{1} = m_{2} a_{2}$ .....(i)

For the motion of the cylinder

$f_{1} + f_{2} = m_{1} a_{1}$ .....(ii)

and $f_{1} R - f_{2} R = I α$ .....(iii)

Also $a_{2} = a_{1} + α R$ ......(iv)

For no slipping of cylinder, $a_{1} = α R$ ......(v)

Solving above equations, we get

$a_{1} = \frac{4 F}{3 m_{1} + 8 m_{2}}, a_{2} = \frac{8 F}{3 m_{1} + 8 m_{2}}$

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