Question

A vertical closed cylinder is separated into two parts by a frictionless piston of mass $m$ and of negligible thickness. The piston is free to move along the length of the cylinder. The length of the cylinder above piston is $l_{1},$ and that below the piston is $l_{2},$ such that $l_{1}>l_{2} .$ Each part of the cylinder contains $n$ moles of an ideal gas at equal temperature $T$. If the piston is stationary, its mass $m$ will be given by: (R is universal gas constant and $g$ is the acceleration due to gravity)

• A. $\frac{R T}{n g}\left[\frac{l_{1}-3 l_{2}}{l_{1} l_{2}}\right]$
• B. $\frac{n R T}{g}\left[\frac{l_{1}-l_{2}}{l_{1} l_{2}}\right]$
• C. $\frac{n R T}{g}\left[\frac{1}{l_{2}}+\frac{1}{l_{1}}\right]$
• D. $\frac{R T}{g l}\left[\frac{2 l_{1}+l_{2}}{l_{1} l_{2}}\right]$

Answer 1

Answer: (B)

$P_{1}+\frac{m g}{A}=P_{2}$
$\Rightarrow \frac{n R T}{l_{1} A}+\frac{m g}{A}=\frac{n R T}{l_{2} A}$
$\Rightarrow m=\frac{n R T}{g}\left(\frac{1}{l_{2}}-\frac{1}{l_{1}}\right)$
$=\frac{n R T}{g}\left(\frac{l_{1}-l_{2}}{l_{1} l_{2}}\right)$