A piston of mass M is hung from a massless… - JEE Main 2025 Physics PYQ

JEE Main 2025 — Physics Mechanics

2025mcqhard

A piston of mass $M$ is hung from a massless spring whose restoring force law goes as $\mathrm{F}=-\mathrm{kx}^3$ , 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 ' $n$ ' 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 $\mathrm{L}_0$ to $\mathrm{L}_1$ , the total energy delivered by the filament is (Assume spring to be in its natural length before heating)

JEE Main 2025 Physics, Work Power Energy — question figure

Official previous-year question

Held on 3 Apr 2025 · Verified 6 Jul 2026.

Options

A
$3 n R T \ln \left(\frac{L_1}{L_0}\right)+2 \mathrm{Mg}\left(L_1-L_0\right)+\frac{k}{3}\left(L_1^3-L_0^3\right)$
B
$n R T \ln \left(\frac{L_1^2}{L_0^2}\right)+\frac{M g}{2}\left(L_1-L_0\right)+\frac{k}{4}\left(L_1^4-L_0^4\right)$
C
$n R T \ln \left(\frac{L_1}{L_0}\right)+M g\left(L_1-L_0\right)+\frac{k}{4}\left(L_1^4-L_0^4\right)$
D
$n R T \ln \left(\frac{L_1}{L_0}\right)+M g\left(L_1-L_0\right)+\frac{3 k}{4}\left(L_1^4-L_0^4\right)$

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Solution

Using WET

Total energy supplied $=$ gravitational potential energy + spring potential energy + work done by gas

$\begin{aligned}
& \operatorname{Mg} \quad\left(L_1-L_0\right)+\int_{L_0}^{L_1} k^3 d x+n R T \ell n \
& {\left[\frac{L_1 \mathrm{A}}{\mathrm{L}0 \mathrm{~A}}\right]+\mathrm{W}{\mathrm{ext}}=0} \
& \frac{\mathrm{~K}}{4}\left[\mathrm{x}^4\right]_{\mathrm{L}_0}^{\mathrm{L}_1}+\operatorname{Mg}\left(\mathrm{L}_1-\mathrm{L}0\right)+\int{\mathrm{L}_0}^{\mathrm{L}_1} k x^3 d x+n R T \ell n
\end{aligned}$

$\begin{aligned} & {\left[\frac{L_1}{L_0}\right]+\mathrm{W}_{\mathrm{ext}}=0} \\ & \frac{\mathrm{k}}{4}\left(\mathrm{~L}_1^4-\mathrm{L}_0^4\right)+\operatorname{Mg} \quad\left(\mathrm{L}_1-\mathrm{L}_0\right)+\mathrm{nRT} \ell \mathrm{n} \\ & {\left[\frac{\mathrm{L}_1}{\mathrm{~L}_0}\right]+\mathrm{W}_{\mathrm{ext}}=0} \\ & \mathrm{~W}_{\mathrm{ext}}=\frac{\mathrm{k}}{4}\left(\mathrm{~L}_1^4-\mathrm{L}_0^4\right)+\operatorname{Mg}\left(\mathrm{L}_1-\mathrm{L}_0\right)+\mathrm{nRT} \ell \mathrm{n}\left[\frac{\mathrm{L}_1}{\mathrm{~L}_0}\right]\end{aligned}$

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