A sealed vertical cylinder of radius \(R\) and height \(h=0.60 \mathrm{~m}\) is initially filled halfway with water, and the upper half is filled with air. The air is initially at standard atmospheric pressure, \(p_{0}=1.01 \cdot 10^{5} \mathrm{~Pa}\). A small valve at the bottom of the cylinder is opened, and water flows out of the cylinder until the reduced pressure of the air in the upper part of the cylinder prevents any further water from escaping. By what distance is the depth of the water lowered? (Assume that the temperature of water and air do not change and that no air leaks into the cylinder.)

The ideal gas law is a fundamental principle that relates the pressure, volume, temperature, and amount (in moles) of an ideal gas. It is represented by the equation

PV = nRT

where P is the pressure, V is the volume, n is the number of moles of the gas, R is the universal gas constant, and T is the temperature in Kelvin. This equation shows that the pressure and volume of a gas are directly proportional to the number of moles and the temperature of the gas. When applying this law to real-life problems, such as determining the change in air pressure in a cylinder, assumptions are often made like the temperature and the amount of gas (in moles) remain constant during the process. In the textbook exercise, the ideal gas law enables us to connect the initial and final states of the air in the cylinder after some water has been drained.

Hydrostatic pressure is the pressure exerted by a fluid at equilibrium at any given point within the fluid, due to the force of gravity. It increases in proportion to depth according to the formula:

\( P = P_0 + \rho gh \)

Here, P is the hydrostatic pressure, P_0 is the external pressure, \( \rho \) is the fluid's density, g is the acceleration due to gravity, and h is the depth of the fluid. This concept is crucial when dealing with the cylinder water air interface in fluid mechanics problems. It allows us to understand how the pressure changes as water is removed from the cylinder, affecting the pressure inside the cylinder and consequently the level of the water.

The cylinder water-air interface refers to the boundary between the water and air within the cylinder. At this interface, hydrostatic balance must be maintained, meaning the pressure inside the air must equal the pressure exerted by the water column plus the atmospheric pressure. Any change in the volume of air within the cylinder, as seen in the exercise, affects the pressure at this interface due to the ideal gas law. In the beginning, with the valve closed, the pressure is uniform, matching the atmospheric pressure. When the valve is open, water leaves the cylinder causing the air to expand and thus its pressure to change, which alters the equilibrium at the water-air interface.

The pressure-volume relationship is also known as Boyle's Law in gas dynamics. It states that the pressure of a gas is inversely proportional to its volume when temperature and mass remain constant.

\( P_1V_1 = P_2V_2 \)

This relationship is a crucial part of the ideal gas law and comes into play when analyzing closed-system problems in fluid mechanics. In the given exercise, as the water level drops and the volume of air increases inside the cylinder, the pressure exerted by the air decreases. By establishing the initial conditions (V1, P1) and relating them to the final conditions (V2, P2), we use this relationship to find the new height to which the water level drops.