A vertical cylinder of crosssectional area A is fitted with...

A vertical cylinder of crosssectional area $A$ is fitted with a tight-fitting, frictionless piston of mass $m$ (Fig. P18.40). The piston is not restricted in its motion in any way and is supported by the gas at pressure $P$ below it. Atmospheric pressure is $P_{0} .$ We wish to find the height $h$ in Figure $\mathrm{P} 18.40 .$ (a) $\mathrm{What}$ analysis model is appropriate to describe the piston? (b) Write an appropriate force equation for the piston from this analysis model in terms of $P, P_{0}, m, A,$ and $g .$ (c) Suppose $n$ moles of an ideal gas are in the cylinder at a temperature of $T$ Substitute for $P$ in your answer to part (b) to find the height $h$ of the piston above the bottom of the cylinder.

Key Concepts

Static Equilibrium

This concept involves analyzing a system in which every force is balanced such that the net force is zero. In this problem, the piston is considered to be in static equilibrium because it is not accelerating. This means that the sum of the forces acting upward (from the gas pressure) is exactly balanced by the downward forces (its weight and the external atmospheric pressure). It forms the basis for modeling the system with a free?body diagram and applying Newton’s first law.

Force Balance Equation

Writing a force balance equation involves identifying all forces acting on the piston and setting their sum equal to zero in the static case. The upward force is provided by the pressure from the gas acting over the area, while the downward forces include the gravitational force on the piston and the atmospheric pressure acting on its top. This approach leads to an equation that relates the gas pressure to the atmospheric pressure, the mass of the piston, its cross-sectional area, and the acceleration due to gravity.

Ideal Gas Law

The ideal gas law (PV = nRT) relates the pressure, volume, and temperature of a given amount of gas. In this problem, once the force balance determines the pressure under the piston, the ideal gas law is invoked to substitute for the gas pressure in terms of the number of moles, the gas temperature, and the volume occupied by the gas. Using the cylinder geometry (volume = cross-sectional area multiplied by height) connects the thermodynamic state of the gas directly to the mechanical position of the piston.

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