Q. 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 $\ell_1$, and that below the piston is $\ell_2$ , such that $\ell_1$>$\ell_2$. 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)

Solution:

$P_2 A \, = \, P_1A \, + \, mg$
$\cfrac{nRT.A}{A\ell_2} \, = \cfrac{nRT.A}{A\ell_2} +mg$
$nRT \bigg(\cfrac{1}{\ell_2} \, - \, \cfrac{1}{\ell_1}\bigg) = mg$
$m \, = \, \cfrac{nRT}{g} \, \bigg(\cfrac{\ell_1\ell_2}{\ell_1 .\ell_2}\bigg)$

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