A piston of mass M is attached to a massless spring whose restoring force is F = -kx³. The piston separates a vertical chamber into two parts, and the lower part contains n moles of an ideal gas. The spring is initially at its natural length. The gas is heated isothermally at temperature T so that the piston rises from height L₀ to L₁. The total energy supplied by the heating filament is:

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G1 Physics · Accepted Answer

At piston position x, the gas pressure must support the piston weight and spring force. Thus the mechanical work associated with raising the piston from L₀ to L₁ contains the gravitational and spring terms. The work against gravity is W g W_g W g ​ = Mg(L₁-L₀). For the nonlinear spring force kx³, W s W_s W s ​ = ∫[L₀ to L₁] kx³ dx = (k/4)(L₁⁴-L₀⁴). Because the gas expansion is isothermal, ΔU = 0. The ideal-gas isothermal expansion term is W i s o W_iso W i ​ so = nRT ln(V₁/V₀). Since the cylind…