Question

The coefficient of friction between the block A of mass m and the triangular wedge B of mass M is $\mu$. There is no friction between the wedge and the plane. If the system (block A + wedge B) is released so that there is no sliding between A and B. Find the inclination $\theta$.

Answer 1

Acceleration of the system (A+B) down the plane = $gsin\theta$. In both the block and wedges, friction between them will be static. The block A is at rest w.r.t. wedge B.
From FBD of A:
$N=mg-masin\theta$ (i)
$f=macos\theta$ (ii)
For no sliding between A and B,
$f<\mu N$
$mgsin\theta .cos\theta <\mu (mg-mgsi{n}^2\theta )$
or $mgsin\theta cos\theta <\mu mg(1-sin\theta )$
$sin\theta cos\theta <\mu co{s}^2\theta$
$\Rightarrow tan\theta$ or $\theta =ta{n}^{-1}\left(\mu \right)$