The coefficient of static friction between the two blocks shown in figure is `mu` and the table is smooth. What maximum horizontal forced F can be applied to he block of mass M so that the block move together?

Explanation

When the maximum force F is applied, bothteh blocks move together towards right. The only horizontal force on the upper block of mass m is tht due to the friction by thelower block of mass M.Hence this force onm should be towads right. The force of friction on M by should be towards left by Newton's third law. As we are talking of the maximum possible force F that can be applied, the friction is limiting and hence `f=muN where N` is the normal force between the blocks.
consider the motion of m. The forces on m are ure.

a. mg downward by the earth (gravity),
b.N upward by the block M(normal force ) and
c. `f=mu N`(friction) towards right by the block M.
In the vertical direction there is no acceleration. This gives
N=mg.........i
IN the horizontal direction, let the acceleration be a then
`muN=ma`
or` mumg=ma`
or, `a=mug`.......ii
Next consider the motion of M ure.

Teh forces on M are
a. Mg downward by the earth (gravity) ltbRgt `N_1` upward by the table (normal force),
c. N downward by m (normal force),
`f=muN` (friction) towards left by m and
e. F (aplied force) by the experimenter. The equation of motion is
`F-muN=Ma`
or, F-mu mg=M mug[Using i. and ii.]`
or, `F=mug(M+m)`.