Two piston/cylinder arrangements, A and B, have their gas...

Two piston/cylinder arrangements, $A$ and $B,$ have their gas chambers connected by a pipe, as shown in Fig. $\mathrm{P} 1.81$. The cross-sectional areas are $A_{A}=$ $75 \mathrm{~cm}^{2}$ and $A_{B}=25 \mathrm{~cm}^{2},$ with the piston mass in $A$ being $m_{A}=25 \mathrm{~kg} .$ Assume an outside pressure of $100 \mathrm{kPa}$ and standard gravitation. Find the mass $m_{B}$ so that none of the pistons have to rest on the bottom.

Key Concepts

Pressure Equilibrium in Connected Systems

When two gas chambers are connected by a pipe, the pressure throughout the connected system becomes uniform under static conditions. This principle ensures that any differences in force acting on the pistons are balanced by adjusting the piston positions until the pressure, and hence the forces acting on each piston, are equal.

Force Balance on Pistons

The equilibrium of each piston is achieved when the upward force due to the internal gas pressure acting over its cross?sectional area is balanced by the downward gravitational force from its mass. This balance prevents the piston from accelerating and serves as the basis for determining necessary mass or pressure adjustments.

Effect of Piston Area on Force

The force exerted by gas pressure on a piston is the product of the pressure and the piston’s area. Different piston areas result in different net forces for the same pressure, which must be considered when designing systems to maintain equilibrium without additional support.

Transcript

00:01 So if we're trying to solve for the mass of piston b such that both piston a and b are balancing at some level above the bottom of these chambers, what we're first going to do is we're going to set up two equations that describe how the forces acting on both pistons are in balance when the pistons are at rest.

00:24 So for chamber a, we have some pressure p in the gas chamber, which is resulting in an upward force on the piston.

00:33 And this is going to be equal to the downwards forces, which are the force of gravity, as well as the force due to atmospheric pressure.

00:50 And we're going to have pretty much the same expression for chamber b, for piston b rather, but this time we'll have the subscripts with the values for chamber b.

01:09 So what we know is that in order to have these two pistons in balance, so at rest, some level above the bottom, we need the pressures in both of these chambers to be equal.

01:22 Since they're connected to each other by this tube, so if the pressures are equal, then there's going to be no flow of gas in between.

01:31 That causes the pistons to move.

01:33 So what we're going to do is we're going to divide both sides of these two expressions by the cross -sectional area so that we just have p on this side.

01:46 And these pressures, again, as we mentioned, are going to be equal.

01:50 So then we can set the expressions that we have here equal to each other.

01:55 So this is the equation that we'll end up getting when we equate the pressures...

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