Key ideas
Fluid mechanics covers liquids and gases at rest and in motion. Pressure grows with depth, objects float by displacing fluid, and a flowing fluid trades pressure against speed.
- Pressure at depth h is p = p₀ + ρgh. It acts equally in all directions and depends on depth, not on the shape of the container.
- The buoyant force equals the weight of the fluid displaced: F_b = ρVg. An object floats when its average density is below the fluid’s.
- For an incompressible fluid the flow rate is constant, so A₁v₁ = A₂v₂ and the fluid speeds up in narrow sections.
- Bernoulli’s equation says that where speed is higher, pressure is lower along a streamline (smooth, frictionless flow).
Key equations
p = p₀ + ρghF_b = ρVgA₁v₁ = A₂v₂p + ½ρv² + ρgh = constantF₁ ÷ A₁ = F₂ ÷ A₂Re = ρvD ÷ μ
Worked example
Water flows at 1.0 m/s in a pipe 4.0 cm across, which narrows to 2.0 cm. The area ratio is (4 ÷ 2)² = 4, so v₂ = 4 × 1.0 = 4.0 m/s. For a horizontal pipe, the pressure drop is ½ρ(v₂² − v₁²) = ½ × 1000 × (16 − 1) = 7500 Pa.
Common mistakes
- Using the ratio of diameters instead of the ratio of areas in the continuity equation.
- Mixing gauge pressure (above the atmosphere) with absolute pressure.
- Thinking buoyancy depends on the object’s own weight. It depends on the volume of fluid displaced.
Practice
Test yourself with practice questions on this topic: Fluids