A vertical cylinder of cross-sectional area $S = 10^{- 3} m^{2}$ contains 1 mol of an ideal monatomic gas under a piston of mass M=10 kg. At a certain instant, a heater is switched on which transmits 3500J of heat per second to the gas. If the entire system is thermally insulated and the pressure of the gas under the piston remains constant, then what is the speed (in m/s) of the piston? Take atmospheric pressure to be $P_{0} = 10^{5} Pa$ .

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answer is 7.

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Detailed Solution

$\begin{array}{l} dQ = dU + dW \\ dW = PdV = PSdx \\ dW = (P_{0} + \frac{Mg}{S}) Sdx \\ dW = (P_{0} S + Mg) dx \end{array}$

$\begin{array}{l} PV = RT \\ \Rightarrow PdV = RdT \Rightarrow PSdx = RdT \\ \Rightarrow dU = \frac{3}{2} RdT = \frac{3}{2} PSdx \\ dU = \frac{3}{2} (P_{0} + \frac{Mg}{S}) Sdx = \frac{3}{2} (P_{0} S + Mg) dx \\ dQ = \frac{5}{2} (P_{0} S + Mg) dx \\ \Rightarrow \frac{dx}{dt} = \frac{2}{5} \frac{dQ / dt}{(P_{0} S + Mg)} \\ \Rightarrow V = 7 m / s \end{array}$