Question

The 1.0-m-tall cylinder shown in FIGURE CP14.71 contains air at a pressure of 1 atm. A very thin, fric- tionless piston of negligible mass is placed at the top of the cylinder, to prevent any air from escaping, then mercury is slowly poured into the cylinder until no more can be added without the cylinder over- f lowing. What is the height h of the column of compressed air?

Answer 1

There are a few steps involved in solving this problem, but the key is to remember that the pressure in the cylinder is constant, and to use the fact that the pressure at any point in a fluid is given by the height of the fluid column above that point.

Let's start by finding the pressure at the bottom of the cylinder, which is the same as the pressure inside the piston. Since the cylinder contains only air at a pressure of 1 atm, the pressure at the bottom of the cylinder is also 1 atm.

Next, we need to find the pressure at the top of the mercury column. We know that the pressure at the top of the cylinder is also 1 atm, since the piston is preventing any air from escaping. The pressure at the top of the mercury column is given by the height of the column, so we need to find the height at which the pressure of the mercury is equal to 1 atm. We can look up the density of mercury (13,600 kg/m^3) and use the formula P = ρgh, where P is the pressure, ρ is the density, g is the acceleration due to gravity (9.81 m/s^2), and h is the height of the fluid column. Solving for h, we get:

h = P / (ρg) = 1 atm / (13600 kg/m^3 × 9.81 m/s^2) ≈ 0.073 m

So the height of the mercury column is 0.073 m.

Finally, we need to find the height of the compressed air column, which is simply the height of the cylinder minus the height of the mercury column:

h = 1.0 m - 0.073 m = 0.927 m

So the height of the compressed air column is 0.927 m.