Mass per unit volume, and what makes fluids different from solids
The Big Idea
Density (ρ = m/V) measures how tightly packed matter is. Water has ρ = 1000 kg/m³. Lead has ρ ≈ 11,300 kg/m³. Air has ρ ≈ 1.2 kg/m³. Fluids (liquids and gases) flow because their molecules aren't locked into a rigid pattern. Liquids hold their volume but flow to fit their container; gases expand to fill any container.
That's the density of aluminum. Since 2700 > 1000, it sinks in water.
Topic 8.2
Pressure
Force per area, and how it changes with depth
The Big Idea
Pressure = Force / Area (P = F/A). Pressure is a scalar — it acts in ALL directions at a point in a fluid. The deeper you go in a fluid, the more weight is above you pushing down. So pressure rises with depth: P = P₀ + ρgh. At the same depth, the pressure is the same everywhere in a connected fluid (Pascal's principle).
Key Equations
Definition
P = F/A. Units: pascals (Pa = N/m²).
Atmospheric
P_atm ≈ 1.0 × 10⁵ Pa at sea level.
Hydrostatic
P = P₀ + ρgh.
Pascal's Principle
Pressure transmits equally through enclosed fluid.
Example
Pressure at the bottom of a pool
A pool is 2.0 m deep. What is the pressure at the bottom? (P_atm = 1.0 × 10⁵ Pa, ρ_water = 1000, g = 10)
P = P₀ + ρgh = 10⁵ + (1000)(10)(2.0) = 10⁵ + 2 × 10⁴ = 1.2 × 10⁵ Pa
That's about 20% above atmospheric pressure. Every 10 m of water depth adds another atmosphere of pressure.
Topic 8.3
Fluids and Newton's Laws (Buoyancy)
Why things float, sink, or hover
The Big Idea
Archimedes' principle: The buoyant force on an object equals the weight of the fluid it displaces. F_b = ρ_fluid · V_displaced · g. This comes from the pressure difference between top and bottom of the object (bottom is deeper, so higher pressure pushes up more than top pushes down). If F_b > weight, the object floats up. If F_b < weight, it sinks. For a floating object, F_b = mg exactly.
Key Facts
Buoyant Force
F_b = ρ_fluid · V_disp · g. Uses FLUID density.
Floating Rule
ρ_object < ρ_fluid → floats.
Sinking Rule
ρ_object > ρ_fluid → sinks.
Fraction Submerged
When floating: ρ_object / ρ_fluid.
Example
A wooden block floating in water
A wooden block has density 600 kg/m³ and volume 1.0 × 10⁻³ m³ (1 liter). What is its mass, and what fraction is submerged when it floats in water?
The block sits 60% under water and 40% above. Check: F_b = (1000)(0.6 × 10⁻³)(10) = 6.0 N = (0.60)(10) = mg ✓
Topic 8.4
Fluids and Conservation Laws (Flow)
Continuity and Bernoulli — mass and energy in moving fluids
The Big Idea
Flowing fluids obey two conservation laws. Continuity (mass conservation): A₁v₁ = A₂v₂. Whatever flows into a pipe must flow out, so a narrower pipe forces faster flow. Bernoulli (energy conservation): P + ½ρv² + ρgh = constant along a streamline. As one term grows, another must shrink. Fast-moving fluid has LOW pressure — that's how airplane wings produce lift.
Key Equations
Continuity
A₁v₁ = A₂v₂. Smaller pipe = faster.
Bernoulli
P + ½ρv² + ρgh = constant.
Flow Rate
Q = A·v. Units: m³/s.
Key Insight
Fast flow = low pressure.
Example
Water through a narrowing pipe
Water flows at 2.0 m/s through a pipe with cross-section 4.0 × 10⁻⁴ m². The pipe narrows to 1.0 × 10⁻⁴ m². What is the water's speed in the narrow section?
The water speeds up by a factor of 4 (matching the 4× area decrease). By Bernoulli, the pressure in the narrow section is LOWER than in the wide section.
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How to use this visual review
Take 2-3 minutes per slide. Read the Big Idea, then the key facts, then the worked example. Four slides total — one per topic.
The two most-tested ideas are buoyancy (8.3) and Bernoulli (8.4). Make sure you can solve a floating-object problem and a pipe-flow problem from start to finish.