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?

Explanation

Choosing the positive direction for x and `varphi` as shown in figure, let us we write the equation of motion for the sphere `F_x=mw_(cx)` and `N_(cx)=I_cbeta_x`
`f r=m_2w_2`, `f_rr=2/5m_2r^2beta`
(`w_2` is the acceleration of the C.M. of sphere.)
For the plank from the Eq. `F_x=mw_x`
`F-f_r=m_1w_1`
In addition, the condition for the absence of slipping of the sphere yields the kinematical relation between the accelerations:
`w_1=w_2+betar`
Simultaneous solution of the four equations yields:
`w_1=(F)/((m_1+2/7m_2))` and `w_2=2/7w_1`