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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 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) \[\frac{RT}{ng}\left[ \frac{{{l}_{1}}-3{{l}_{2}}}{{{l}_{1}}{{l}_{2}}} \right]\]   

B) \[\frac{RT}{g}\left[ \frac{2{{l}_{1}}+{{l}_{2}}}{{{l}_{1}}{{l}_{2}}} \right]\]

C) \[\frac{nRT}{ng}\left[ \frac{1}{{{l}_{2}}}+\frac{1}{{{l}_{2}}} \right]\]  

D) \[\frac{nRT}{g}\left[ \frac{{{l}_{1}}-{{l}_{2}}}{{{l}_{1}}{{l}_{2}}} \right]\]

Correct Answer: D

Solution :

\[{{P}_{2}}A-{{P}_{1}}A=mg\]

\[m=\frac{1}{g}\left( \frac{{{P}_{1}}A{{\ell }_{1}}}{{{\ell }_{1}}}-\frac{{{P}_{2}}A{{\ell }_{2}}}{{{\ell }_{2}}} \right)\]

\[m=\frac{1}{g}\left( \frac{nRT}{{{\ell }_{1}}}-\frac{nRT}{{{\ell }_{2}}} \right).\]