Fluid pressure increases with depth, so the force on a submerged object is not usually found by multiplying one pressure by one area. Calculus lets us add up many thin horizontal strips, each with its own pressure. This is important for designing dams, aquarium walls, submarine windows, and storage tanks.
The main idea is that pressure depends on depth while area depends on the shape of the plate.
Key Facts
- Fluid pressure at depth h is P = rho g h.
- Hydrostatic force on a flat surface is F = integral P dA.
- For a vertical plate, use a thin horizontal strip with dA = w(y) dy.
- If y measures depth below the water surface, then F = integral rho g y w(y) dy.
- The integration limits must match the top and bottom depths of the submerged plate.
- Use rho = 1000 kg/m^3 and g = 9.8 m/s^2 for water near Earth unless stated otherwise.
Vocabulary
- Hydrostatic pressure
- Hydrostatic pressure is the pressure exerted by a fluid at rest due to the weight of the fluid above a point.
- Hydrostatic force
- Hydrostatic force is the total force a fluid at rest exerts on a submerged surface.
- Depth
- Depth is the vertical distance below the fluid surface, and it determines the pressure in a stationary fluid.
- Density
- Density is mass per unit volume and is represented by rho in fluid pressure formulas.
- Differential area
- A differential area is a very small piece of a surface, such as a thin horizontal strip with area dA = w(y) dy.
Common Mistakes to Avoid
- Using the area of the whole plate with the pressure at the top, which is wrong because pressure changes with depth and must be integrated over the plate.
- Measuring y upward from the bottom but using P = rho g y as if y were depth, which gives incorrect pressure values unless the coordinate system is converted correctly.
- Forgetting the strip width w(y), which is wrong because the area of each horizontal strip depends on the shape and width of the plate at that depth.
- Using inconsistent units, which is wrong because rho, g, depth, width, and force must be in compatible units such as kg/m^3, m/s^2, m, and newtons.
Practice Questions
- 1 A vertical rectangular plate is 2 m wide and 3 m tall, with its top edge at the water surface. Using rho = 1000 kg/m^3 and g = 9.8 m/s^2, find the hydrostatic force on one side of the plate.
- 2 A vertical rectangular plate is 4 m wide and 2 m tall, with its top edge 1 m below the water surface. Set up and evaluate the integral for the hydrostatic force on one side of the plate using rho = 1000 kg/m^3 and g = 9.8 m/s^2.
- 3 A triangular plate and a rectangular plate have the same area and extend from the water surface to the same maximum depth. Explain why their hydrostatic forces may be different.