Question

In the pulley arrangement shown, the pulley $P_2$ is movable. Assuming coefficient of friction between $m$ and surface to be $\mu$, the minimum value of $M$ for which $m$ is at rest is

• A. $M=\frac{\mu m}{2}$
• B. $m=\frac{\mu M}{2}$
• C. $M=\frac{m}{2\mu }$
• D. $m=\frac{M}{2\mu }$

Answer 1

The correct option is A $M=\frac{\mu m}{2}$


On displacing $m$ through $x$ towards right, $M$ gets displaced downward by $2x$.
For equilbrium
$T_1=\mu \text{mg}......\left(1\right)$
$T_1=2T_2.....\left(2\right)$
$T_2=\text{Mg}........\left(3\right)$
From equation (1), (2) and (3)
$\Rightarrow 2\left(\text{Mg}\right)=\mu \text{mg}$
$\Rightarrow M=\frac{1}{2}\mu \text{m}$