Density · Pressure · Buoyancy · Flow · 10–15% of the AP Physics 1 exam
ρ = m/V
P = P₀ + ρgh
F_b = ρVg
A₁v₁ = A₂v₂
P + ½ρv² + ρgh = const
The basics
What's covered: Density, pressure (including hydrostatic), buoyancy and Archimedes' principle, fluid flow (continuity), and Bernoulli's equation.
Exam weight: 10–15% — one of the larger units. Don't skip it.
The big trick: Bernoulli's equation is just energy conservation per volume. Continuity is just mass conservation. Both are familiar physics in fluid clothing.
Use g = 10 m/s² and ρ_water = 1000 kg/m³ for most AP problems.
📐 Key equations
ρ = m/V
Density. Mass per unit volume. Units: kg/m³. Water = 1000.
P = F/A
Pressure. Force per area. Units: Pa = N/m². Scalar — acts in all directions.
P = P₀ + ρgh
Hydrostatic pressure. Pressure at depth h below a fluid surface at pressure P₀.
F_b = ρ_fluid · V_disp · g
Buoyant force (Archimedes). Equal to weight of displaced fluid. Use the FLUID's density.
F_b = mg (floating)
Floating condition. For a floating object, buoyant force equals its weight.
f_sub = ρ_object / ρ_fluid
Fraction submerged. When floating, the fraction of the object below the surface equals the density ratio.
W_apparent = mg − F_b
Apparent weight. What a scale reads underwater. True weight minus buoyancy.
Bernoulli's equation. Sum of pressure + KE per volume + gravitational PE per volume is constant along a streamline.
Q = A·v
Volume flow rate. Volume of fluid passing per second. Units: m³/s.
The 4 topics at a glance
8.1 Structure & Density
ρ = m/V. Definition of density. Specific gravity = density ratio to water. Distinguishing solids, liquids, and gases.
8.2 Pressure
P = F/A; P = P₀ + ρgh. Pressure increases with depth. Pascal's principle: applied pressure transmits through enclosed fluids.
8.3 Fluids and Newton's Laws
F_b = ρ_fluid · V_disp · g. Archimedes' principle. Floating, sinking, apparent weight. Free-body diagrams in fluids.
8.4 Conservation Laws
A₁v₁ = A₂v₂ (mass); P + ½ρv² + ρgh = const (energy). Continuity and Bernoulli. Fast flow = low pressure.
🛟 How to solve a buoyancy problem (3 steps)
1. Draw a free-body diagram on the object: weight (mg, down) and buoyant force (F_b, up). For submerged or sinking objects, you may also have a normal force or tension.
2. Write F_b = ρ_fluid · V_displaced · g. For floating: V_displaced = submerged volume. For fully submerged: V_displaced = object's total volume.
3. Apply Newton's second law. For floating or hovering: F_b = mg. For sinking or rising: F_b − mg = ma.
💨 How to solve a Bernoulli problem (3 steps)
1. Identify two points along a streamline. Usually one where you know everything, and one where you don't.
2. Write Bernoulli at both points. P₁ + ½ρv₁² + ρgh₁ = P₂ + ½ρv₂² + ρgh₂.
3. If needed, combine with continuity (A₁v₁ = A₂v₂) to eliminate one velocity. Then solve.
⚠️ Common exam traps
The buoyant force formula uses the FLUID's density, not the object's. ρ_fluid is what matters. The object's density only enters through whether it floats, and how much is submerged.
For floating objects, fraction submerged = density ratio. An object with half the density of water sits exactly half-submerged.
Pressure depends only on depth, not on container shape. A skinny tall column and a wide short one give the same pressure at the same depth.
Bernoulli says faster flow = LOWER pressure. Counterintuitive — many students get the sign wrong.
Continuity assumes incompressible flow. Works for liquids (always) and for gases at slow speeds (often).
Pressure is a SCALAR. It doesn't have a direction. The FORCE from pressure on a surface does have a direction (perpendicular to the surface, into it).
Use absolute pressure in Bernoulli's equation — gauge pressure cancels if it appears on both sides, but be consistent.