A block of mass m is placed on a wedge of mass M and a force F is acting on it as shown . If all surface are smooth , find the force F such that , there is no relative motion between m and M .

Explanation

Since block and wedge both move together , they can be assumed as a system .
,
Let a be the common acceleration
` :. F = (M + m ) a ` … (A)
Now draw FBD of block m
In inertial f.o.r (Ground )

` N sin theta = ma` ...(i)
`N cos theta = mg ` ...(ii)
From (i) and (ii)
` a = tan theta `
Putting the value of a in equation (A)
` F = (M +m)g tan theta`
In Non - inertial f.o.r. (Wedge)

` sum F_(xx)= 0`
` rArr N sin theta - ma = 0 ` ....(i)
`sum F_(y) = 0 rArr Ncos theta - mg = 0 ` ....(ii) '
This yields the same set of equations and the same result i.e F = `(M + m) g tan theta `