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?

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

$a_p=a_c + \alpha r \dots \left(1\right)$

Here $r$ is the radius of the sphere

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

$f=m_2{a}_c\dots \left(2\right)$

and $f r =I\alpha$

or $f r =\frac{2}{5}{m}_2{r}^2\alpha \dots \left(3\right)$

For the motion of the plank

$F-f=m_1{a}_p\dots \left(4\right)$

After solving the equation,

$a_c=\left[\frac{2F}{2m_2+7m_1}\right]$

and $a_p=\left[\frac{7F}{2m_2+7m_1}\right]$