Question 47 of 183
A piston of mass is hung from a massless spring whose restoring force law goes as , where k is the spring constant of appropriate dimension. The piston separates the vertical chamber into two parts, where the bottom part is filled with ' ' moles of an ideal gas. An external work is done on the gas isothermally (at a constant temperature T) with the help of a heating filament (with negligible volume) mounted in lower part of the chamber, so that the piston goes up from a height to , the total energy delivered by the filament is (Assume spring to be in its natural length before heating)
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Answer: (C)
Using WET
Total energy supplied gravitational potential energy + spring potential energy + work done by gas
$\begin{aligned}
& \operatorname{Mg} \quad\left(L_1-L_0\right)+\int_{L_0}^{L_1} k^3 d x+n R T \ell n \\
& {\left[\frac{L_1 \mathrm{~A}}{\mathrm{~L}_0 \mathrm{~A}}\right]+\mathrm{W}_{\mathrm{ext}}=0} \\
& \frac{\mathrm{~K}}{4}\left[\mathrm{x}^4\right]_{\mathrm{L}_0}^{\mathrm{L}_1}+\operatorname{Mg}\left(\mathrm{L}_1-\mathrm{L}_0\right)+\int_{\mathrm{L}_0}^{\mathrm{L}_1} k x^3 d x+n R T \ell n
\end{aligned}$
Total energy supplied gravitational potential energy + spring potential energy + work done by gas
$\begin{aligned}
& \operatorname{Mg} \quad\left(L_1-L_0\right)+\int_{L_0}^{L_1} k^3 d x+n R T \ell n \\
& {\left[\frac{L_1 \mathrm{~A}}{\mathrm{~L}_0 \mathrm{~A}}\right]+\mathrm{W}_{\mathrm{ext}}=0} \\
& \frac{\mathrm{~K}}{4}\left[\mathrm{x}^4\right]_{\mathrm{L}_0}^{\mathrm{L}_1}+\operatorname{Mg}\left(\mathrm{L}_1-\mathrm{L}_0\right)+\int_{\mathrm{L}_0}^{\mathrm{L}_1} k x^3 d x+n R T \ell n
\end{aligned}$
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