Question

Two identical block $A$ and $B$ are placed on a rough inclined plane of inclination ${45}^o$. The coefficient of friction between block $A$ and incline is $0.2$ and that between $B$ and incline is $0.3$. The initial separation between the two blocks is $\sqrt{2}m$. The two blocks are released from rest, then find
$\left(a\right)$ The time after which front faces of both blocks come in same line and
$\left(b\right)$ The distance moved by each block for attaining above position.

Answer 1

From, $mg\sin \theta -\mu N=ma$; where $N=mg\cos \theta$
$a_A=g\sin {45}^o-0.2g\cos {45}^o=4\sqrt{2}m/s^2$
$a_B=g\sin {45}^o-0.3g\cos {45}^o=\frac{7}{2}\sqrt{2}m/s^2$
Relative acceleration
$a_{AB}=0.5\sqrt{2}m/s^2$
Relative displacement
$s_{AB}=\frac{1}{2}{a}_{AB}{t}^2=\sqrt{2}$(given)
$t^2=\frac{2\sqrt{2}}{0.5\sqrt{2}}=4\Rightarrow t=2sec$.
$a_B=\frac{1}{2}{a}_B{t}^2=7\sqrt{2}m$
$s_A=\frac{1}{2}{a}_A{t}^2=8\sqrt{2}m$