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 body diagrams of the sphere and the plank are as shown below :
Writing equations of motion :
For sphere : Linear acceleration
`a_(1) = (mu_(s) Mg)/(M) = mu_(s) g `
Angular acceleration
`alpha = ((mu_(s) Mg) R)/((2)/(5) MR^(2))=(5)/(2)(mu_(s)g)/(R)" ". . . (ii)` For plank : Linear acceleartion
`a_(2) = (F - mu_(s) Mg)/(m) " ". . . (iii)`
For no slipping acceleration of point B and A is same,
So: `a_(2) = a_(1) + Ralpha`
Solving the above four equation, we get
`F = mu_(s) g(M + (7)/(2)m) `
Thus, maximum value of F can be `mu_(s) g (M + (7)/(2) m) `