Determine the maximum horizontal force `F` that may be applied to the plank of mass `m` for which the solid sphere does not slip as it begins to roll on the plank. The sphere has a mass `M` and radius `R`. The coefficient of static and kinetic friction between the sphere and the plank are `mu_(S)` and `mu_(k)` respectively.

Explanation

the free diagrams of the sphere and the plank are as shown below:
Writing equation of motion
For sphere equations of motion
For sphere linear acceleration `a_(1)=(mu_(s)Mg)/(M)=mu_(s)g` .. (i)
Angular acceleration `alpha=((mu_(s)Mg)R)/((2)/(5)MR^(2))`
`=(5)/(2)(mu_(S)g)/(R)` ...(ii)
For plank linear acceleration
`a_(2)=(F-mu_(S)Mg)/(m)` ..(iii)
For no slipping `a_(2)=a_(1)+Ralpha` ..(iv)
Solving the above four equations we get
`F=mu_(S)g(M+(7)/(2)m)`
Thus, maximum value of `F` can be
`mu_(S)g(M+(7)/(2)m)`