In the arrangement shown in figure, cylinder is thermally insulated. An ideal gas is filled in the cylinder having pressure P0 (> atmospheric pressure Pa ). Spring of force constant k is initially unstretched. Piston of mass m and area S is frictionless. In equilibrium piston rises up a distance x0, then:

Engineering

Physics

Thermodynamics

Question

In the arrangement shown in figure, cylinder is thermally insulated. An ideal gas is filled in the cylinder having pressure P₀ (> atmospheric pressure P_a ). Spring of force constant k is initially unstretched. Piston of mass m and area S is frictionless. In equilibrium piston rises up a distance x₀, then:

A

Final pressure of the gas is $P_{a} + \frac{{kx}_{0}}{S} + \frac{mg}{S}$

B

Decrease in internal energy of the gas is $\frac{1}{2} {kx}_{0}^{2} + mg x_{0} + P_{a} {Sx}_{0}$

C

Work done by the gas is $\frac{1}{2} {kx}_{0}^{2} + {mgx}_{0}$

D

Work done by the gas is 6kx₀²

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Final pressure = P_a + $\frac{{kx}_{0}}{S} + \frac{mg}{S} (\frac{net force}{Area})$

Work done by gas $\int P dv$

$= \int_{0}^{x_{0}} PS dx$

$= \int_{0}^{x_{0}} (P_{a} + \frac{kx}{S} + \frac{mg}{S}) S dx$

$= P_{a} {Sx}_{0} + {mgx}_{0} + \frac{{kx}_{0}^{2}}{2}$

$∵ ∆ Q = 0$ (thermally instated)

$∆ U = ∆ W$

$∴ ΔU = (P_{a} {Sx}_{0} + {mgx}_{0} + \frac{{kx}_{0}^{2}}{2})$