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Q.

A thermally insulated vertical cylinder of cross-section area A contains 2 moles of an ideal monatomic gas under a piston of mass M which can move without friction. A heater supplying heat to the gas at constant rate P J/s , is switched on at $t = 0$ . The velocity of the piston is given by [Take atmospheric pressure as $p_{0}$ ]

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a

$\frac{2 P}{5 [p_{0} A + M g]}$

b

$\frac{3 P}{5 [p_{0} A + M g]}$

c

$\frac{3 P}{2 [p_{0} A + M g]}$

d

$\frac{P}{p_{0} A + m g}$

answer is A.

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Detailed Solution

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

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