AP Physics 2 Exam Question:

A cylindrical container is filled with a liquid of density ρ. The container has a height of h and a radius of r. Consider a small section of the liquid at a depth y from the top of the container. Calculate the pressure at this depth in terms of ρ, g, and y, and derive the equation for the total pressure at the bottom of the container.

Answer:

To calculate the pressure at a depth y, we can use the equation for the pressure of a fluid:

P = P₀ + ρgh

Where:

• P: pressure at the depth y
• P₀: pressure at the surface of the liquid (usually atmospheric pressure)
• ρ: density of the liquid
• g: acceleration due to gravity
• h: height of the liquid column above the depth y

To derive the equation for the total pressure at the bottom of the container, we consider the pressure at the bottom of the container (P_b) which is equal to the pressure at the surface (P₀) plus the pressure due to the height of the liquid column (h).

At the bottom of the container, the height of the liquid column (h) is equal to the height of the container (h). Therefore, the equation for the total pressure at the bottom of the container is:

P_b = P₀ + ρgh

Let's calculate the pressure at a depth y and derive the equation for the total pressure at the bottom of the container step-by-step.

1. Calculate the pressure at a depth y:

Given:

• Density of the liquid (ρ) = ρ
• Acceleration due to gravity (g) = g
• Depth from the top of the container (y) = y

Using the equation for pressure of a fluid:

P = P₀ + ρgh

At the depth y, the height of the liquid column (h) is equal to the total height of the container (h) minus the depth y:

h = h - y

Substituting this value into the equation for pressure:

P = P₀ + ρg(h - y)

Simplifying:

P = P₀ + ρgh - ρgy

Final equation for the pressure at depth y:

P = P₀ + ρgh - ρgy

2. Derive the equation for the total pressure at the bottom of the container:

At the bottom of the container, the depth (y) is equal to the height of the container (h). Therefore, substituting this value into the equation for pressure at a depth y:

P_b = P₀ + ρg(h - h) = P₀

The pressure at the surface (P₀) is usually equal to the atmospheric pressure, so the equation for the total pressure at the bottom of the container is:

P_b = P₀

Therefore, the total pressure at the bottom of the container does not depend on the depth or height of the container, only on the atmospheric pressure.

This completes the calculation of the pressure at a depth y and the derivation of the equation for the total pressure at the bottom of the container.