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 the 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 an equal temperature $T$. If the piston is stationary, its mass $m$ will be given by:
($R$ is universal gas constant and $g$ is acceleration due to gravity)
(A) $\dfrac{{nRT}}{g}\left[ {\dfrac{1}{{{l_2}}} + \dfrac{1}{{{l_1}}}} \right]$
(B) $\dfrac{{nRT}}{g}\left[ {\dfrac{{{l_1} - {l_2}}}{{{l_1}{l_2}}}} \right]$
(C) $\dfrac{{RT}}{g}\left[ {\dfrac{{2{l_1} + {l_2}}}{{{l_1}{l_2}}}} \right]$
(D) $\dfrac{{RT}}{g}\left[ {\dfrac{{{l_1} - 3{l_2}}}{{{l_1}{l_2}}}} \right]$

Answer 1

Hint
Since the piston is stationary, the forces on it will be balanced. So we can equate the mass above and below the piston as well as the mass of the piston. So from there we can find the value of the mass of the piston.
In this solution we will be using the following formula,
$\Rightarrow F = PA$
Where $F$ is the force, $P$ is the pressure and $A$ is the area.
$\Rightarrow PV = nRT$
Where $V$ is volume, $n$ is number of moles, $R$ is universal gas constant and $T$ is temperature.

Complete step by step answer
In this problem the piston in the cylinder is said to be stationary. So the amount of net force on the piston has to be zero. Now the forces working on the piston are the mass of the gas above the piston and its own weight in the downward direction. And the amount of gas below the piston exerts a force on it in the upward direction.
For the calculation let us consider the surface area of the base of the cylinder to be $A$. So the surface area of the piston also has to be $A$ to fit perfectly in the cylinder.
Let us denote the pressure on the piston due to the gas above it is given by ${P_1}$. From the equation of ideal gas, we can write,
$\Rightarrow {P_1}{V_1} = nRT$ where $n$ and $T$ are constants for both the parts. The volume in the region above the piston is given by the product of the surface area and the length. The length of the upper part is given in the question as, ${l_1}$. Therefore, ${V_1} = {l_1}A$
So we can write,
$\Rightarrow {P_1} = \dfrac{{nRT}}{{{V_1}}} = \dfrac{{nRT}}{{{l_1}A}}$
The force on the piston due this gas downwards is,
$\Rightarrow {F_1} = {P_1}A$
Then, substituting we get
$\Rightarrow {F_1} = \dfrac{{nRT}}{{{l_1}A}}A$
Cancelling the $A$,
$\Rightarrow {F_1} = \dfrac{{nRT}}{{{l_1}}}$
Similarly, the pressure on the piston due to the gas below it is given by ${P_2}$. From the equation of ideal gas, we can write,
$\Rightarrow {P_2}{V_2} = nRT$. The volume in the region below the piston is given by the product of the surface area and the length. The length of the lower part is given in the question as, ${l_2}$. Therefore, ${V_2} = {l_2}A$
So we can write,
$\Rightarrow {P_2} = \dfrac{{nRT}}{{{V_2}}} = \dfrac{{nRT}}{{{l_2}A}}$
The force on the piston due this gas upwards is,
$\Rightarrow {F_2} = {P_2}A$
Then, substituting we get
$\Rightarrow {F_2} = \dfrac{{nRT}}{{{l_2}A}}A$
Cancelling the $A$,
$\Rightarrow {F_2} = \dfrac{{nRT}}{{{l_2}}}$
The mass of the piston is $m$ so the force acting on it downwards due to its mass is, $mg$
So the equation for the piston will be,
$\Rightarrow {F_1} + mg = {F_2}$
Substituting we get,
$\Rightarrow \dfrac{{nRT}}{{{l_1}}} + mg = \dfrac{{nRT}}{{{l_2}}}$
Taking $\dfrac{{nRT}}{{{l_1}}}$ from LHS to the RHS we get
$\Rightarrow mg = \dfrac{{nRT}}{{{l_2}}} - \dfrac{{nRT}}{{{l_1}}}$
We can take $nRT$ as common, so
$\Rightarrow mg = nRT\left( {\dfrac{1}{{{l_2}}} - \dfrac{1}{{{l_1}}}} \right)$
So the mass will be,
$\Rightarrow m = \dfrac{{nRT}}{g}\left( {\dfrac{1}{{{l_2}}} - \dfrac{1}{{{l_1}}}} \right)$
Taking LCM as ${l_2}{l_1}$,
$\Rightarrow m = \dfrac{{nRT}}{g}\left( {\dfrac{{{l_1} - {l_2}}}{{{l_1}{l_2}}}} \right)$
Hence the correct answer is option (B).

Note
According to Newton's second law, the net force on a body produces acceleration on the body. Since in this case there is no acceleration on the body as it is stationary, for the sum of the forces that are acting on the body will be zero.