AP Physics 2 Exam Question

A container is filled with a non-viscous fluid of density ρ. The container has a horizontal section with a cross-sectional area A1 on the left, and a vertical section with a cross-sectional area A2 on the right. The fluid surface is at a height h above the top of the vertical section. The container is open to the atmosphere.

a) Derive an equation for the pressure at the bottom of the vertical section of the container in terms of the given variables.

b) A small hole is then opened at the bottom of the vertical section of the container. Determine the speed at which the fluid emerges from the hole.

Assume that the fluid is incompressible and that there is no friction or losses due to the hole.

Answer:

a) To find the pressure at the bottom of the vertical section, we can use the equation for fluid pressure:

ΔP = ρgh

where ΔP is the pressure difference, ρ is the density of the fluid, g is the acceleration due to gravity, and h is the height difference.

In this case, the pressure at the bottom of the vertical section is the same as the atmospheric pressure since the container is open to the atmosphere. Therefore, the pressure at the bottom (P2) can be written as:

P2 = Patm + ΔP

where Patm is the atmospheric pressure.

From the equation for fluid pressure, we can rewrite ΔP as:

ΔP = ρgh

Substituting this expression into the previous equation, we have:

P2 = Patm + ρgh

This is the equation for the pressure at the bottom of the vertical section of the container.

b) To determine the speed at which the fluid emerges from the hole, we can use Torricelli's law, which relates the speed of efflux of a fluid through a small hole at the bottom of a container to the height of the fluid column above the hole.

The equation for Torricelli's law is:

v = √(2gh)

where v is the speed of efflux, g is the acceleration due to gravity, and h is the height difference.

In this case, the height difference h is equal to the height of the fluid surface above the hole. Therefore, the speed of efflux can be written as:

v = √(2gh)

Substituting the values of g and h into this equation, we have:

v = √(2 * 9.8 m/s^2 * h)

Simplifying further, we get:

v = √(19.6h)

This is the equation for the speed at which the fluid emerges from the hole.