Question

A fixed thermally conducting cylinder has a radius $R$ and height $L_0$. The cylinder is open at its bottom and has a small hole at its top. A piston of mass $M$ is held at a distance $L$ from the top surface as shown in Fig. The atmospheric pressure is $p_0$.
While the piston is at a distance $2L$ from the top, the hole at the top is sealed. The piston is then released to a position where it can stay in equilibrium. In this condition, the distance of the piston from the top is

• A. $\left(\frac{2p_0\pi R^2}{\pi R^2{p}_0+Mg}\right)\left(2L\right)$
• B. $\left(\frac{p_0\pi R^2-Mg}{\pi R^2{p}_0}\right)\left(2L\right)$
• C. $\left(\frac{p_0\pi R^2+Mg}{\pi R^2{p}_0}\right)\left(2L\right)$
• D. $\left(\frac{2p_0\pi R^2}{\pi R^2{p}_0-Mg}\right)\left(2L\right)$

Answer 1

The correct option is D $\left(\frac{2p_0\pi R^2}{\pi R^2{p}_0-Mg}\right)\left(2L\right)$

Let $p$ be the pressure in equilibrium, then $pA=p_{0}A-Mg$
$p=p_0-\frac{Mg}{A}=p_0-\frac{Mg}{\pi R^2}$
Applying $p_1{V}_1=p_2{V}_2$
$L^{\prime }=\frac{2p_{0}L}{p}=\left(\frac{p_0}{p_0-\frac{Mg}{\pi R^2}}\right)\left(2L\right)$
$\left(\frac{p_0\pi R^2}{\pi R^2{p}_0-Mg}\right)\left(2L\right)$
Therefore, option (d) is correct.