Question

A pulley-block system is as shown in the figure. Blocks $2\text{kg}$ and $3\text{kg}$ are at rest. Coefficient of friction between $2\text{kg}$ and incline is

• A. $\frac{2}{\sqrt{3}}$
• B. $\frac{1}{2}$
• C. $\frac{1}{3}$
• D. $\frac{1}{\sqrt{3}}$

Answer 1

The correct option is A $\frac{2}{\sqrt{3}}$

Since the $3\text{kg}$ block tends to pull down due to gravity, $2\text{kg}$ block tends to move up. Hence, friction on $2\text{kg}$ will be down the plane.
FBD of $2\text{kg}$ block:


As system is at rest, From FBD we can deduce that,
$N=2g\cos \theta .....\left(1\right)$
$T=2g\sin \theta +\mu N.......\left(2\right)$
From $\left(1\right)$ and $\left(2\right)$ we get ,
$T=2g\sin \theta +\mu \left(2g\cos \theta \right).......\left(3\right)$
FBD of $3\text{kg}$ block:

From the figure we can deduce that , $T=3g.......\left(4\right)$
From $\left(3\right)$ and $\left(4\right)$ we get ,
$3g=2g(\sin \theta +\mu \cos \theta )$
$\Rightarrow \frac{3}{2}=\sin {30}^{\circ }+\mu \cos {30}^{\circ }$
$\Rightarrow \mu =\frac{2}{\sqrt{3}}$
Thus, option (a) is the correct answer.