There are two block as shown in the figure of masses `1kg` and `4kg`. Friction coefficient between any two surfaces are `0.2` then find maximum value of horizontal force `F` so that both blocks moves together.

Explanation

a. If both blocks move together, i,e no sliding between `M` and `m`, the friction between `m` and `M` will be static nature .Static friction force is a self - adjusting force `0 le f le mu_(s)N` .Acceleration of system in this case, `a = (F)/((M + m))` ....(i)

`FBD` of `m` and `M`
Equation of motion of `m`
` f = ma = m (F)/((M + m))` ...(ii)
If there is no sliding between `M` and `m` , then `f le f_(s - max)`
`rArr (mF)/((M + m)) le mu_(s)(mg)`
`rArr F le mu_(s)(m)g`
b. If `F gt mu_(s)(M + m)g`, then there will be relative sliding between `M` and `m` . when relative sliding between `M` and `m` static

The friction force reaches limiting value the friction for because `f_(k) = mu_(k)N`
Free - body diagrams of `m` and `M` .
Equation of motion of `m : mu_(k)(mg) = ma_(1)`
Acceleration of `m , a_(1) = mu_(k)g`
Equation of motion of `M: F - mu_(k) mg = Ma_(2)`
Acceleration of `M , a_(2) = (F - mu_(k) mg )/(M)`