Let as consider the next case in the previous illustration when there is no friction between ground and `M` . The coefficient of static and kinetic friction are `mu_(2)` and `mu_(1)`, respectively, and force `F` acts upper block as shown in figure
a. What is the maximum possible value of `F` so that the system moves together ?
b . If there is releative sliding between `M` and `m` then calculate acceleration of `M` and `m`

Explanation

Let the system move together , then `a = (F)/((M + m))`
`FBD` of `m` and `M`
From `FBD` of `M : f = ma = M((F)/(M + m))` ….(i)

As there is no sliding between `M` and `m`
` M((F)/(M + m))le mu_(s) mg`
`rArr F le mu_(s)(m)/(M)(M + m)g`
If `F gt mu_(s)(m)/(M)(M + m)g` the relative sliding between the blocks starts . The friction force between the block will be of kinetic nature and both blocks will move with different accelerations.

Acceleration of `m. a_(1) = (F - mu_(s) mg)/(m)`
Acceleration of `M, a_(2) = (mu_(s) mg)/(M)`