Six grams of air is contained in the cylinder shown in Fig....

Six grams of air is contained in the cylinder shown in Fig. 3.13. The air is heated until the piston raises $50 \mathrm{~mm}$. The spring just touches the piston initially. Calculate $(a)$ the temperature when the piston leaves the stops and $(b)$ the work done by the air on the piston. (a) The pressure in the air when the piston just raises from the stops is found by balancing the forces on the piston: The temperature is found from the ideal-gas law: $$ T=\frac{P V}{m R}=\frac{(193.7)(0.15)(\pi)(0.2)^{2} / 4}{(0.006)(0.287)}=530 \mathrm{~K} \quad \text { or } \quad 257^{\circ} \mathrm{C} $$ (b) The work done by the air is considered to be composed of two parts: the work to raise the piston and the work to compress the spring. The work required to raise the piston a distance of $0.05 \mathrm{~m}$ is $$ W=(F)(d)=(P)(A)(d)=(193.7) \frac{\pi(0.2)^{2}}{4}(0.05)=0.304 \mathrm{~kJ} $$ The work required to compress the spring is $W=\frac{1}{2} K x^{2}=\frac{1}{2}(400)\left(0.05^{2}\right)=0.5 \mathrm{~kJ}$. The total work required by the air to raise the piston is $$ W=0.304+0.5=0.804 \mathrm{~kJ} $$

Six grams of air is contained in the cylinder shown in Fig. 3.13. The air is heated until the piston raises $50 \mathrm{~mm}$. The spring just touches the piston initially. Calculate $(a)$ the temperature when the piston leaves the stops and $(b)$ the work done by the air on the piston.
(a) The pressure in the air when the piston just raises from the stops is found by balancing the forces on the piston:
The temperature is found from the ideal-gas law:
$$
T=\frac{P V}{m R}=\frac{(193.7)(0.15)(\pi)(0.2)^{2} / 4}{(0.006)(0.287)}=530 \mathrm{~K} \quad \text { or } \quad 257^{\circ} \mathrm{C}
$$
(b) The work done by the air is considered to be composed of two parts: the work to raise the piston and the work to compress the spring. The work required to raise the piston a distance of $0.05 \mathrm{~m}$ is
$$
W=(F)(d)=(P)(A)(d)=(193.7) \frac{\pi(0.2)^{2}}{4}(0.05)=0.304 \mathrm{~kJ}
$$
The work required to compress the spring is $W=\frac{1}{2} K x^{2}=\frac{1}{2}(400)\left(0.05^{2}\right)=0.5 \mathrm{~kJ}$. The total work required by the air to raise the piston is
$$
W=0.304+0.5=0.804 \mathrm{~kJ}
$$

Key Concepts

Ideal Gas Law

This principle relates the pressure, volume, temperature, and mass of a gas through a constant specific gas constant. It is essential in determining how a change in one state variable, such as temperature, affects the others, and is widely used in problems concerning the behavior of gases under various conditions.

Force Equilibrium

Force equilibrium involves balancing all forces acting on a system to determine conditions of static or dynamic balance, such as when a piston begins to move. In thermodynamic systems, understanding force balance is crucial for analyzing how pressure forces from a gas counteract external forces like spring resistance or gravitational forces.

Work in Thermodynamic Processes

Work in a thermodynamic context is the energy transferred by a system during expansion or compression against an external force. It often requires integrating variable pressures over a change in volume, and it represents the mechanical energy output or input associated with the system's changes.

Spring Energy and Compression Work

Spring compression work is the energy stored in a spring when it is deformed. It is calculated using the spring constant and the displacement, and it is important in combined mechanical systems where elastic forces contribute to the total energy change during processes such as piston movement.

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