Question

A mass of m is sliding on a inclined plane of mass M which is kept on smooth surface find displacement of mass M.

Answer 1

let the angle of inclination be $\theta$

As clearly seen from the figure,

then the net force acting on mass m in downward direction is

f= $mgsin\theta-f_s$

Where $f_s$ is frictional force on inclined surface

According to Newton's Law Third law,

This force also equal to the force on the inclined plane in upward direction as shown in figure

f=$mgsin\theta-u_smgcos\theta$

Let the horizontal force on Mass M be F

then f=F cos$\theta$

$Fcos\theta=mg(sin\theta-u_scos\theta)$

$F=mg(tan\theta-u_s)$ ....(1)

Let the acelaration of complete system be a

( m +M ) a=F

a=$\frac{F}{(m+M)}$

Displacement in time t is given by second equation of motion as

s=$ut+\frac{1}{2}at^2$

u=0(initial velocity is 0)

=$\frac{1}{2}\times\frac{F}{m+M}\times t^2$

=$\frac{mg(tan\theta-u_s)t^2}{2(m+M)}$