First slide

Application of Newtons laws

Question

A particle of mass m is placed on the smooth face of an inclined plane of mas s M and slope $θ$ which is free to slide on a smooth horizontal plane in a direction perpendicular to the edge. If the particle slides with an acceleration a, the acceleration of the inclined plane towards right A will be

Difficult

A

$\frac{m a \cos θ}{(M + m)}$
B

$\frac{M m \cos θ}{(M + m)}$
C

$\frac{M a \cos θ}{(M + m)}$
D

$\frac{M \cos θ}{(M + m)}$

Solution

As the particle moves down the inclined plane, let a be its acceleration. Now the inclined plane moves horizontally towards right with acceleration A.
Considering the vertical component of forces acting on the particle.
$m g - R \cos θ = m a \sin θ . . . . . . . (i)$
For horizontal components the motion of the particle is considered at the accelerated frame. So a horizontal fictious force acts opposite to A. This pseudo force mA acts towards the left.
$\begin{array}{l} R \sin θ + m A = m a \cos θ \\ R \sin θ = m (a \cos θ - A) . . . . . . (i i) \end{array}$
Now consider the force on inclined plane. The forces are shown in figure.
$\begin{array}{l} R \sin θ = M A . . . . . (i i i) \\ R \cos θ + M g = S . . . . . . . . . (i v) \end{array}$
$From equations (ii) and (iii), we get A = \frac{m a \cos θ}{(M + m)}$

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