Let it roll

A plank of mass \(M\) is moving on a smooth horizontal surface with speed \(v_{\circ}\). At time \(t=0\), a sphere of mass \(m\) and radius \(r\) is gently placed on it and simultaneously. A constant horizontal force \(F\) is applied on the plank in the opposite direction of \(v_{\circ}\). Find the time at which sphere starts pure rolling on the plank . The coefficient of friction between the plank and sphere is \(\mu\).


\(M=4~kg\), \(m=2~kg\), \(\mu=0.6\), \(g=10~m/s^2\),\(v_{\circ}=5~ m/s\), \(F=40~N\).

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