In the arrangement shown in Fig. 12.17, gas is thermally...

In the arrangement shown in Fig. $12.17$, gas is thermally insulated. An ideal gas is filled in the cylinder having pressure $P$ greater than atmospheric pressure $P_{0}$. The spring of force constant $K$ is initially upstretched. The piston of mass $m$ and area s is frictionless. In equilibrium, the piston rises up a distance $x_{0}$, then (A) Final pressure of the gas is $P_{0}+\frac{K x_{0}}{s}+\frac{m g}{s}$. (B) Work done by the gas is $\frac{1}{2} K x_{0}^{2}+m g x_{0}$. (C) Decrease in internal energy of the gas is $\frac{1}{2} K x_{0}^{2}+m g x_{0}+P_{0} s x_{0}$ (D) All of the above.

In the arrangement shown in Fig. $12.17$, gas is thermally insulated. An ideal gas is filled in the cylinder having pressure $P$ greater than atmospheric pressure $P_{0}$. The spring of force constant $K$ is initially upstretched. The piston of mass $m$ and area s is frictionless. In equilibrium, the piston rises up a distance $x_{0}$, then
(A) Final pressure of the gas is $P_{0}+\frac{K x_{0}}{s}+\frac{m g}{s}$.
(B) Work done by the gas is $\frac{1}{2} K x_{0}^{2}+m g x_{0}$.
(C) Decrease in internal energy of the gas is $\frac{1}{2} K x_{0}^{2}+m g x_{0}+P_{0} s x_{0}$
(D) All of the above.

Key Concepts

Mechanical Equilibrium and Force Balance

This concept concerns analyzing the forces acting on a system—in this case, a piston under the influence of gas pressure, atmospheric pressure, spring force, and gravitational force. It dictates that at equilibrium, the net force acting on the piston is zero. This principle is used to derive relationships between the gas pressure and the other forces (such as the additional force due to the spring and the weight of the piston), leading to expressions that balance the system's forces.

Work Done by a Thermodynamic System

In thermodynamics, when a gas expands or contracts, it performs work on its surroundings. The gas does work by displacing a piston against external forces, such as atmospheric pressure, gravity, or a spring. The amount of work done is calculated by integrating the external force over the displacement. This concept explains the energy transferred from the gas as it expands and how it is allocated to different forms of potential energy.

Energy Conservation in Adiabatic Processes

When a system is thermally insulated, it undergoes an adiabatic process where no heat is exchanged with the surroundings. Under these conditions, the first law of thermodynamics implies that any change in the internal energy of the gas is equal to the work done by the gas. This principle is crucial when determining how the decrease in the internal energy of the gas corresponds to the work done against mechanical forces, such as gravity, the spring, and atmospheric pressure.

Spring and Gravitational Potential Energy

As the piston moves, energy is stored in the spring as elastic potential energy and in the gravitational field as gravitational potential energy. The work done by the gas not only overcomes atmospheric pressure but also contributes to these potential energy changes. Understanding how energy is partitioned into different forms (like ½ Kx² for the spring and mgx for gravitational potential energy) is essential for analyzing the overall energy budget in mechanical systems coupled with thermodynamic processes.

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