A thermally insulated vertical cylinder of cross-section area A contains 3 moles of an ideal monatomic gas under a piston of mass M. A heater supplying heat to the gas at constant rate P $\frac{J}{s}$ is switched on at $t = 0$ . The velocity of the piston is given by $(t a k e P_{a t m} = P_{0})$ .

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$\frac{2 P}{5 (P_{0} A + M g)}$

$\frac{3 P}{5 (P_{0} A + M g)}$

$\frac{3 P}{2 (P_{0} A + M g)}$

$\frac{P}{P_{0} A + M g}$

answer is A.

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Detailed Solution

$\begin{array}{l} \frac{d w}{d Q} = 1 - \frac{d U}{d Q} (∴ d Q = d U + d w) \\ = 1 - \frac{1}{γ} \\ = 1 - \frac{1}{(\frac{5}{3})} = 1 - \frac{3}{5} = \frac{2}{5} \\ \Rightarrow \frac{d w}{d Q} = \frac{2}{5} \\ d w = \frac{2}{5} d Q (∴ d Q = P d t) \\ d w = \frac{2}{5} P d t (∴ d w = F d s) \\ F d s = \frac{2}{5} P d t \\ (P_{0} A + M g) \frac{d s}{d t} = \frac{2}{5} P \\ V = \frac{2 P}{5 (P_{0} A + M g)} \end{array}$