This AP Physics 1 lesson on fluid mechanics notes gives the key ideas first, then worked examples and diagrams. After that, practise with the worksheet on fluid mechanics and check yourself with the quiz.
Quick answer: Fluids obey a few linked rules: density is mass per volume, pressure rises with depth as P = P₀ + ρgh, buoyant force equals the weight of fluid displaced, continuity says Av is constant in a pipe, and Bernoulli shows that faster flow means lower pressure.
Level: AP Physics 1 (algebra-based) · Curriculum: AP P1 Unit 8
Fluid mechanics notes: the key ideas
What are density and pressure?
Density tells you how much mass is packed into a given volume. A litre of cooking oil has less mass than a litre of water, which is why oil spreads over the top of a pan of water. Pressure tells you how a force is spread over a surface. Liquids and gases are both fluids, and in a fluid at rest the pressure pushes on every surface it touches, always at right angles to that surface.
Density is mass per unit volume, ρ = m/V, measured in kg/m³. Pressure is the force acting perpendicular to a surface divided by the area of that surface, P = F/A, measured in pascals (1 Pa = 1 N/m²).
| Material | Density (kg/m³) |
|---|---|
| Air (room temperature) | 1.2 |
| Cooking oil | 920 |
| Ice | 917 |
| Fresh water | 1000 |
| Seawater | 1025 |
| Aluminium | 2700 |
| Mercury | 13600 |
How does pressure change with depth?
Picture the water above Chiamaka as she dives in a lake near Lagos. Every layer above her adds weight, so the deeper she goes, the more the fluid above pushes down. The pressure also includes the push of the atmosphere on the surface. The result is a simple straight-line law that depends only on depth, not on the width or shape of the container.
In a fluid of constant density at rest, the absolute pressure at depth h is P = P₀ + ρgh. The term ρgh is the gauge pressure, and P₀ is the pressure at the surface (1.01 × 10⁵ Pa for open air at sea level).
Chiamaka dives to 12 m in fresh water. Find the absolute pressure.
Data: ρ = 1000 kg/m³, g = 9.81 m/s², P₀ = 1.01 × 10⁵ Pa. ρgh = 1000 × 9.81 × 12 = 1.18 × 10⁵ Pa. P = 1.01 × 10⁵ + 1.18 × 10⁵ = 2.19 × 10⁵ Pa, about twice atmospheric pressure.
What is the buoyant force?
Pressure grows with depth, so the water pushes up on the bottom of a submerged object harder than it pushes down on the top. The difference is an upward force. Archimedes found that this force equals the weight of the fluid the object pushes aside.
FB = ρfluid Vsub g
FB is the buoyant force (N), ρfluid the fluid density (kg/m³), Vsub the submerged volume (m³) and g = 9.81 m/s². An object floats when ρobject is less than ρfluid.
Wanjiru places a block of mass 0.60 kg and volume 1.0 × 10⁻³ m³ in water. Does it float, and how much is submerged?
ρ = 0.60 / 1.0 × 10⁻³ = 600 kg/m³, below 1000, so it floats. At rest FB = W = 0.60 × 9.81 = 5.9 N. Then Vsub = 5.886 / (1000 × 9.81) = 6.0 × 10⁻⁴ m³, so 60 % of the block is under water.
What does the continuity equation say?
Water is almost incompressible, so the volume that enters one end of a pipe each second must leave the other end. Where the pipe is narrow, the water has to move faster. This is why your thumb over a hose makes the jet shoot out.
Continuity: A₁v₁ = A₂v₂. The volume flow rate Q = Av (in m³/s) is the same at every section of a pipe of steady, incompressible flow.
Aarav’s main pipe has a diameter of 4.0 cm and carries water at 0.80 m/s. It narrows to 2.0 cm. Find the new speed.
Area scales with diameter squared, so A₁/A₂ = (4.0/2.0)² = 4. Then v₂ = 4 × 0.80 = 3.2 m/s. The flow rate is Q = π(0.020)² × 0.80 = 1.0 × 10⁻³ m³/s, which is 1.0 litre per second.
How does Bernoulli’s equation link speed and pressure?
Bernoulli’s equation is energy conservation for a flowing fluid. Along a streamline, pressure energy, kinetic energy and gravitational energy per unit volume trade off, so faster fluid at the same height has lower pressure.
P + ½ρv² + ρgy = constant along a streamline, for steady, incompressible flow without friction.
Water at 2.0 m/s and 1.80 × 10⁵ Pa enters a horizontal pipe section where it speeds up to 6.0 m/s. Find the new pressure.
Same height, so P₂ = P₁ + ½ρ(v₁² − v₂²) = 1.80 × 10⁵ + 500 × (4.0 − 36) = 1.80 × 10⁵ − 1.6 × 10⁴ = 1.64 × 10⁵ Pa.
Do not use the buoyant force formula with the whole volume of a floating object. Only the submerged part pushes water aside. Also remember that pressure depends on depth, never on how wide the container is.
Before any fluid problem, ask which law applies: depth (P₀ + ρgh), buoyancy (displaced weight), same flow rate (A v), or speed change (Bernoulli). Name it first, then write the equation.
- Density ρ = m/V in kg/m³; pressure P = F/A in pascals
- Pressure at depth: P = P₀ + ρgh, with P₀ = 1.01 × 10⁵ Pa at sea level
- Buoyant force F_B = ρ_fluid V_sub g; it floats when the object’s density is lower
- Continuity for incompressible flow: A₁v₁ = A₂v₂
- Bernoulli: P + ½ρv² + ρgy is constant along a streamline
Density, pressure, buoyancy and moving fluids: frequently asked questions
What is the difference between gauge pressure and absolute pressure?
Gauge pressure is the pressure above atmospheric pressure, ρgh in a liquid. Absolute pressure adds the atmospheric pressure at the surface, so P = P₀ + ρgh. A tyre gauge reads gauge pressure.
Why does a steel ship float when a steel nail sinks?
A hull encloses a large volume of air, so the ship’s average density is below that of water. It pushes aside enough water to equal its weight, while a solid nail cannot.
Does water pressure depend on the shape of the container?
No. At a given depth the pressure depends only on the depth, the fluid density and the surface pressure. A narrow tube and a wide lake give the same pressure at the same depth.
Why does water speed up in a narrower pipe?
Water is nearly incompressible, so the volume per second must be the same everywhere. A smaller area therefore needs a larger speed, following A₁v₁ = A₂v₂.
When can Bernoulli’s equation be used?
Use it for steady flow of an incompressible fluid with negligible friction, along one streamline. It fails when the flow is turbulent, when energy is added by a pump or when friction is large.
