Fluids
Fluids closes out the course by applying force and energy ideas to a substance that deforms rather than staying rigid. Pressure-vs-depth and Pascal's principle handle fluids at rest; Archimedes' principle (a force-balance argument, structurally identical to Unit 2's equilibrium problems) explains buoyancy; and the continuity equation plus Bernoulli's equation -- essentially conservation of mass and energy applied to a moving fluid -- handle fluids in motion.
What you'll learn
- Calculate pressure in a fluid as a function of depth.
- Apply Pascal's principle to hydraulic systems.
- Apply Archimedes' principle to find buoyant force and predict floating/sinking.
- Calculate density and relate it to buoyancy and pressure.
- Apply the continuity equation to relate flow speed and cross-sectional area.
- Apply Bernoulli's equation to relate pressure and speed in flowing fluids.
- Distinguish static fluid problems (pressure, buoyancy) from dynamic ones (flow, continuity).
1. Density and Pressure
Density is mass per unit volume, \(\rho = m/V\). Pressure is force per unit area, \(P = F/A\) — a scalar that, in a static fluid, acts equally in every direction at a given point (not just "downward" like weight).
2. Pressure as a Function of Depth
In a static fluid, pressure increases linearly with depth: \(P = P_0 + \rho gh\), where \(P_0\) is the pressure at the surface (often atmospheric pressure) and \(h\) is the depth below that surface. This is why pressure builds noticeably as you dive deeper underwater, and why it depends only on depth and the fluid's density — not on the container's shape or width.
3. Pascal's Principle and Hydraulic Systems
Pascal's principle states that pressure applied anywhere to an enclosed, incompressible fluid is transmitted equally throughout the fluid. This is the basis for hydraulic lifts: since the pressure is the same at both pistons, \(P = F_1/A_1 = F_2/A_2\), a small force on a small piston can produce a much larger force on a large piston — a form of mechanical advantage.
4. Archimedes' Principle and Buoyancy
Archimedes' principle: the buoyant force on a submerged (or floating) object equals the weight of the fluid it displaces: \(F_B = \rho_{fluid} V_{displaced}\, g\). An object floats if its average density is less than or equal to the fluid's density — in that case, it settles until it displaces just enough fluid for the buoyant force to equal its own weight. An object denser than the fluid sinks, since even displacing its *entire* volume isn't enough buoyant force to support it.
5. The Continuity Equation
For an incompressible fluid flowing through a pipe, the volume flow rate must be the same everywhere along the pipe: \(A_1v_1 = A_2v_2\). A narrower cross-section forces the fluid to speed up to keep the same amount flowing past every point per second.
6. Bernoulli's Equation
Bernoulli's equation is a statement of energy conservation for a flowing fluid: \(P_1 + \tfrac{1}{2}\rho v_1^2 + \rho gh_1 = P_2 + \tfrac{1}{2}\rho v_2^2 + \rho gh_2\). For horizontal flow (\(h_1 = h_2\)), the height terms cancel, revealing a direct trade-off between speed and pressure: faster-moving fluid has *lower* pressure. This is the principle behind lift over an airplane wing and the Venturi effect.
Key equations
- P = F/A — Basic pressure calculation
- P = P0 + ρgh — Pressure as a function of depth
- FB = ρfluid V g — Buoyancy (Archimedes' principle)
- A1v1 = A2v2 — Continuity equation
- P1 + 1/2ρv1^2 + ρgh1 = P2 + 1/2ρv2^2 + ρgh2 — Bernoulli's equation