In the adjoining figure, the coefficient of friction between wedge ( of mass M) and block ( of mass m) is `mu`.
Find the minimum horizontal force F rquired to keep the block stationary with respect to wedge

Explanation

Such problems can be solved with or without using the concepte of pseudo force. Let us, solve the problem by both the methods.
a= Acceleration of ( wedge+ block ) in horizontal direction `= (F)/(M+m).`
(i) Inertial frame of reference ( ground) FBD of block with respect to ground ( only real forces have to be applied) with respect to ground block is moving with an acceleration a.

Therefore,
` sumF_(y)=0 " and " sum F_(x)=ma`
`mg= muN " and " N=ma`
`therefore a=(g)/(mu)`
`therefore F= (M+m)a=(M+m)(g)/(mu)`
(ii) Non- inertial frame of reference ( wedge ) FBD of m with respect to wedge ( real + one pseudo force ) with respect to wedge block is stationary.

`therefore sumF_(x)=0=sumF_(y)`
`therefore mg=muN`
and N= ma
`therefore a=(g)/(mu)` and `F=(M+m)a=(M+m)(g)/(mu)`
From the above discussion, we can see that from both the methods results are same.