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 $L_0$ to $L_1$, the total energy delivered by the filament is : (Assume spring to be in its natural length before heating)
$nRT \ln \left( \frac{L_1}{L_0} \right) + Mg(L_1 - L_0) + \frac{k}{4} (L_1^4 - L_0^4)$
The problem involves determining the total energy delivered by a heating filament that causes a piston to move upward in a vertical chamber containing an ideal gas. The spring obeys a non-linear restoring force law, and the change occurs isothermally.
Approach:
- Gas Work: For an isothermal process, the work done on the gas can be expressed as: \(W_{\text{gas}} = nRT \ln \left( \frac{V_1}{V_0} \right)\), where \(V_0 = A L_0\) and \(V_1 = A L_1\) are the initial and final volumes respectively. Thus, \(W_{\text{gas}} = nRT \ln \left( \frac{L_1}{L_0} \right)\).
- Gravitational Work: The work done against gravity as the piston moves from height \(L_0\) to \(L_1\) is given by: \(W_{\text{gravity}} = Mg (L_1 - L_0)\).
- Spring Work: For a spring force given by \(F = -kx^3\), the potential energy stored in the spring when the displacement changes from \(L_0\) to \(L_1\) is: \(W_{\text{spring}} = \int_{L_0}^{L_1} kx^3 \, dx = \frac{k}{4} (L_1^4 - L_0^4)\).
Summing these works, the total energy delivered by the filament is:
W_{\text{total}} = nRT \ln \left( \frac{L_1}{L_0} \right) + Mg (L_1 - L_0) + \frac{k}{4} (L_1^4 - L_0^4)
This matches the correct option given:
\(nRT \ln \left( \frac{L_1}{L_0} \right) + Mg(L_1 - L_0) + \frac{k}{4} (L_1^4 - L_0^4)\)
Thus, the correct answer is:
Correct Answer: \(nRT \ln \left( \frac{L_1}{L_0} \right) + Mg(L_1 - L_0) + \frac{k}{4} (L_1^4 - L_0^4)\)
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