💧 Density and Pressure

A fluid is anything that flows — gases and liquids. AP Physics 1 (Unit 8) treats fluids as incompressible and non-viscous unless told otherwise.

Density

$\rho = \frac{m}{V}$

• $\rho$ = density (kg/m³)
• $m$ = mass (kg)
• $V$ = volume (m³)

Reference: $\rho_{\text{water}} = 1000$ kg/m³ = 1 g/cm³.

Relative (specific) density = $\rho_{\text{object}}/\rho_{\text{water}}$. If it's < 1 the object floats in water; > 1 it sinks.

Pressure

$P = \frac{F_\perp}{A}$

• $P$ = pressure (Pa = N/m²)
• $F_\perp$ = force perpendicular to the surface
• $A$ = area

Pressure is a scalar — it acts in all directions at a point.

Hydrostatic (Gauge) Pressure

For a fluid at rest, the pressure increases linearly with depth:

$P = P_0 + \rho g h$

• $P_0$ = pressure at the surface (often atmospheric, $\approx 1.0\times10^{5}$ Pa)
• $\rho g h$ is called the gauge pressure — pressure above atmospheric

💡 Key Idea: Pressure depends only on depth and fluid density, not on container shape or total volume.

Pascal's Principle

A pressure change applied to an enclosed, incompressible fluid is transmitted undiminished to every point in the fluid.

For a hydraulic lift:

$\frac{F_1}{A_1} = \frac{F_2}{A_2}$

A small force on a small piston produces a much larger force on a large piston (mechanical advantage = $A_2/A_1$).

⚠️ Common Mistakes

1. Using container shape: Pressure at depth $h$ does not depend on container shape — only $h$.
2. Confusing absolute vs. gauge pressure: $P_{\text{abs}} = P_{\text{atm}} + \rho g h$.
3. Mixing units: 1 L = 0.001 m³. Always convert to SI.

Key Formulas