A block of mass `m` lies on wedge of mass `M` , which lies on fixed horizontal surface. The wedge is free to move on the horizontal surface. A horizontal force of magnitude `F` is applied on block as shown neglecting friction at all surface, the value of force `F` such that block has no relative motion w.r.t wedge will be: (Where `g` is acceleration due to gravity)

Options

A. `(M+m)g tantheta`
B. `(M+m)g cottheta`
C. `(m)/(M)(M+m)g tantheta`
D. `(m)/(M)(M+m)g costheta`

Answer

C

Explanation

For no relative motion between wedge and block, let the acceleration of both block and wedge be `'a'` toward left.
From FBD of block, `Ncostheta=mg` (i)
and `F-Nsintheta=ma` (ii)
From FBD of wedge, `Nsintheta=Ma` (iii)
Solving equation (i), (ii) gand (iii), we get
`F=(m)/(M)(M+m)gtantheta` .