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

A vertical cylinder of crosssectional area $A$ is fitted with a tightfitting, frictionless piston of mass $m$ (Fig. $P 16.56) .$ 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 P16.56. (a) 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

Mechanical Equilibrium

This concept involves analyzing a system in which the net force equals zero, meaning all forces acting on the object balance out. In the context of the piston, mechanical equilibrium allows us to set the upward force from the gas pressure equal to the sum of the downward forces, including atmospheric pressure and the weight of the piston, establishing a condition from which other parameters can be determined.

Free-Body Diagram

A free-body diagram is a simplified representation of an object where all the external forces acting on it are depicted. For the piston, the free-body diagram would include the upward force due to the gas pressure acting on its bottom surface, the downward force from the atmospheric pressure on its top surface, and the gravitational force (its weight). This visualization is crucial to developing the correct force balance equation.

Newton's Second Law and Force Balance

Newton’s Second Law states that the sum of the forces on an object is equal to its mass times its acceleration. In static problems such as the piston in equilibrium, the acceleration is zero, which leads to a force balance. This approach allows us to equate the net upward force to the net downward force, forming an equation that includes pressure forces and the weight of the piston.

Ideal Gas Law

The ideal gas law, stated as PV = nRT, relates the pressure, volume, temperature, and the number of moles of a gas. In this problem, it provides a way to substitute the gas pressure in terms of the measurable quantities like the number of moles, temperature, and volume. This substitution is essential for determining the height of the piston by linking the macroscopic characteristics of the gas to the force balance in the piston setup.

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