JEE Main · 2025 · 03 Apr 2025 · Shift 1 · Heat & Thermodynamics

A piston of mass M is hung from a massless spring whose restoring force law goes as

F = - 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

L_{0}

L_{1}

, the total energy delivered by the filament is (Assume spring to be in its natural length before heating)

(A) $3 n R T ln (\frac{L _{1}}{L _{0}}) + 2 Mg (L_{1} - L_{0}) + \frac{k}{3} (L_{1}^{3} - L_{0}^{3})$
(B) $n R T ln (\frac{L _{1}^{2}}{L _{0}^{2}}) + \frac{M g}{2} (L_{1} - L_{0}) + \frac{k}{4} (L_{1}^{4} - L_{0}^{4})$
(C) $n R T ln (\frac{L _{1}}{L _{0}}) + M g (L_{1} - L_{0}) + \frac{k}{4} (L_{1}^{4} - L_{0}^{4})$ ✓ Correct
(D) $n R T ln (\frac{L _{1}}{L _{0}}) + M g (L_{1} - L_{0}) + \frac{3 k}{4} (L_{1}^{4} - L_{0}^{4})$

Answer: (C)

Solution

The energy supplied by the filament is shared among the work done by the gas, the rise in gravitational potential energy of the piston, and the energy stored in the spring. In the isothermal expansion the gas does work

W_{g a s} = n R T ln \frac{V _{1}}{V _{0}} = n R T ln \frac{L _{1}}{L _{0}}

, since the volume is proportional to the column length. Raising the piston of mass

M

through

(L_{1} - L_{0})

needs

M g (L_{1} - L_{0})

. The spring, obeying

F = k x^{3}

, stores

\int_{L_{0}}^{L_{1}} k x^{3} d x = \frac{k}{4} (L_{1}^{4} - L_{0}^{4}) .

Adding these contributions, the total energy delivered is

E = n R T ln (\frac{L _{1}}{L _{0}}) + M g (L_{1} - L_{0}) + \frac{k}{4} (L_{1}^{4} - L_{0}^{4}) .

JEE Main · 2025 · 03 Apr 2025 · Shift 1 · Single correct option

Practise more: all JEE Main PYQs on Heat & Thermodynamics →

JEE Main · 2025 · 03 Apr 2025 · Shift 1 · Heat & Thermodynamics

Solution

More JEE Main questions on Heat & Thermodynamics