A plank of mass $m_{1}$ with a uniform sphere of mass $m_{2}$ placed on it rests on a smooth horizontal plane. A constant horizontal force $F$ is applied to the plank. With what acceleration will the plank and the centre of the sphere move, provided there is no sliding between the plank and the sphere?

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$a_{p} = [\frac{F}{2 m_{2} + 7 m_{1}}]$

$a_{p} = [\frac{8 F}{2 m_{2} + 7 m_{1}}]$

$a_{p} = [\frac{7 F}{2 m_{2} + 7 m_{1}}]$

$a_{p} = [\frac{7 F}{m_{2} + 7 m_{1}}]$

answer is B.

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Detailed Solution

If $a_{c}$ is the acceleration of the centre of the centre of the sphere and $a_{p}$ is the acceleration of the plank, then

$a_{p} = a_{c} + α r \dots (1)$

Here $r$ is the radius of the sphere

For the motion of sphere: If $f$ is the frictional force, then

$f = m_{2} a_{c} \dots (2)$

and $f r = I α$

or $f r = \frac{2}{5} m_{2} r^{2} α \dots (3)$

For the motion of the plank

$F - f = m_{1} a_{p} \dots (4)$

After solving the equation,

$a_{c} = [\frac{2 F}{2 m_{2} + 7 m_{1}}]$

and $a_{p} = [\frac{7 F}{2 m_{2} + 7 m_{1}}]$

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