Fluids, Pressure, and Buoyancy

By the end of this lesson, you will be able to calculate fluid pressure, apply Pascal's Principle, use Archimedes' Principle for buoyancy, and apply the continuity equation and Bernoulli's equation.

Fluid Pressure

Pressure: force per unit area.

$$P = \frac{F}{A}$$

Units: Pascal (Pa) = N/m².

Pressure in a fluid at depth $h$:

$$P = P_0 + \rho gh$$

$P_0$ = surface pressure; $\rho$ = fluid density; $g = 9.8$ m/s².

Pascal's Principle

Pressure applied to a confined fluid is transmitted equally in all directions.

Hydraulic systems: $\frac{F_1}{A_1} = \frac{F_2}{A_2}$

A small force on a small piston creates a large force on a large piston (same pressure, different area). Used in hydraulic jacks, brakes, and elevators.

Archimedes' Principle

Buoyant force = weight of fluid displaced.

$$F_b = \rho_{\text{fluid}} V_{\text{displaced}} g$$

Object floats if: $F_b \geq W$, i.e., $\rho_{\text{object}} \leq \rho_{\text{fluid}}$.

Object sinks if: $\rho_{\text{object}} > \rho_{\text{fluid}}$.

Example 1 — Buoyancy

Will a wooden block ($\rho = 600$ kg/m³, volume 0.1 m³) float in water?

$F_b = 1000 \times 0.1 \times 9.8 = 980$ N; $W = 600 \times 0.1 \times 9.8 = 588$ N

$F_b > W$ → floats; 60% submerged (until $F_b = W$).

Fluid Flow: Continuity and Bernoulli

Continuity equation (incompressible fluid):

$$A_1 v_1 = A_2 v_2$$

Flow rate is constant; fluid speeds up in narrower pipes.

Bernoulli's Equation: energy conservation for flowing fluids.

$$P_1 + \frac{1}{2}\rho v_1^2 + \rho g h_1 = P_2 + \frac{1}{2}\rho v_2^2 + \rho g h_2$$

Bernoulli's principle: faster flow → lower pressure (explains airplane lift, curveball in baseball).

Summary

Pressure: $P = F/A$; increases with depth ($P = P_0 + \rho gh$).

Pascal: pressure transmitted equally → hydraulic force multiplication.

Archimedes: $F_b = \rho_f V_d g$; float when $\rho_{\text{object}} < \rho_{\text{fluid}}$.

Continuity: $A_1v_1 = A_2v_2$; faster in narrow sections.

Bernoulli: fast flow → low pressure.