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Difficulty: MediumHydrostatic Pressure

A hydroelectric dam holds back a massive artificial lake. Along the submerged face of the dam, there is a small diagnostic sensor port and a massive turbine water intake. Both the sensor port and the center of the turbine intake are located exactly 45 meters45\text{ meters} below the surface of the lake. The diameter of the turbine intake is twenty times larger than the diagnostic port. When the water in the lake is perfectly still, how does the hydrostatic pressure at the diagnostic port compare to the pressure at the center of the turbine intake?

  1. The pressure is identical at both locations because they are positioned at the exact same depth.Answer
  2. B
    The pressure is twenty times greater at the turbine intake because its larger diameter captures more force from the water.
  3. C
    The pressure is significantly higher at the diagnostic port because the fluid is compressed into a narrower opening.
  4. D
    The pressure at both locations is determined by the total mass of the reservoir, so the depth of the specific openings is not the primary factor.

Answer

The pressure is identical at both locations because they are positioned at the exact same depth.
Hydrostatic pressure in a static fluid is calculated using the formula P=ρghP = \rho g h (Pressure = density × gravity × depth). Because both the diagnostic port and the center of the turbine intake are located exactly 45 meters45\text{ meters} below the surface, they experience the exact same water pressure. The size, diameter, or area of the opening affects the total force exerted on that opening, but it does not change the pressure, which is a measure of force per unit area.

Step-by-Step Solution

1
Identify the factors that determine hydrostatic pressure in a static fluid.
Hydrostatic pressure (PP) is determined by the formula P=ρghP = \rho g h, where ρ\rho is fluid density, gg is gravity, and hh is depth.
This establishes the physical rule governing the scenario, showing that only depth matters when the fluid is consistent.
2
Compare the depth of the two locations provided in the scenario.
Both the diagnostic port and the turbine intake center are located at a depth of 45 meters45\text{ meters}.
Since the hh value is identical for both locations, the pressure calculated from P=ρghP = \rho g h will be equal.
3
Evaluate the effect of the opening sizes on the hydrostatic pressure.
The diameter of the openings has no effect on the hydrostatic pressure, though it does affect the total force (F=P×AF = P \times A).
It separates the concept of pressure (force per unit area) from the concept of total force, confirming that the pressure remains identical.

Key Concept

Hydrostatic pressure in a liquid depends only on the depth of the liquid and its density, and is entirely independent of the shape or size of the container or opening.
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Hydrostatic Pressure

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