Fluids

What this unit covers

The topics below follow the published Physics 1 course framework for Unit 8. This unit is worth 10–15% of the exam, so budget your time against that rather than against how long the unit takes to teach.

Lessons in this unit

• Density & Pressure13 min · 3 objectivesCompute density as ρ = m/V · Define pressure as force per unit area, P = F/A · Find how pressure increases with depth, P = ρgh
• Buoyancy13 min · 3 objectivesState Archimedes’ principle · Compute the buoyant force, F_b = ρ_fluid·V_displaced·g · Predict floating versus sinking from a density comparison
• The Continuity Equation12 min · 3 objectivesState the continuity equation for incompressible flow · Relate cross-sectional area to flow speed, A₁v₁ = A₂v₂ · Predict how flow speed changes in pipes of varying width
• Bernoulli’s Equation14 min · 3 objectivesState Bernoulli’s principle relating speed and pressure · Apply Bernoulli’s equation along a streamline · Explain everyday lift effects with the speed–pressure trade-off
• Floating, Sinking & Apparent Weight15 min · 3 objectivesDerive the fraction of a floating object that is submerged from a density ratio · Compute apparent weight for a submerged object as a force balance · Predict how a fluid's density changes whether an object floats, sinks or hovers
• Bernoulli as Energy Conservation15 min · 3 objectivesRead each term in Bernoulli's equation as an energy per unit volume · Combine continuity and Bernoulli to analyze flow through a changing pipe · State the assumptions behind Bernoulli's equation and identify where they fail

Formulas in Unit 8

Every term in Unit 8

All 45 terms we publish for Fluids, with definitions. Reading them through is the fastest way to find the ones you cannot define — then drill those in cram mode until you can produce them without the prompt.

Pressure with depth

P = P₀ + ρgh. It depends on depth, not on the shape or total volume of the container.

Floating condition

A floating object displaces fluid weighing exactly its own weight, so the fraction submerged equals the ratio of the densities.

Bernoulli's equation

P + ½ρv² + ρgh is constant along a streamline for ideal flow, so faster-moving fluid is at lower pressure.

Applying Bernoulli and continuity together

Continuity gives the speed change from geometry; Bernoulli converts that speed change into a pressure difference.

Why pressure does not depend on container shape

Pressure depends only on depth, fluid density and surface pressure, so a narrow tube and a wide tank at the same depth read the same.

Why ships float

A steel hull encloses a large volume of air, so the average density of the ship is below that of water even though steel is far denser.

Flow rate

Volume per unit time, Av. Conserved along a pipe for an incompressible fluid regardless of how the cross-section changes.

Limits of Bernoulli's equation

Assumes steady, incompressible, non-viscous flow along a streamline. Real flows with turbulence or viscosity depart from it.

Density

ρ = m/V, in kg/m³. Determines whether an object floats in a given fluid, independently of its size.

Pressure

P = F/A, force per unit area in pascals. A scalar — pressure has no direction, though the force it exerts on a surface is perpendicular to it.

Pressure at depth

P = P₀ + ρgh. Depends on depth, not on the shape or total volume of the container — a narrow tube and a wide tank read the same at the same depth.

Gauge vs absolute pressure

Gauge pressure is the amount above atmospheric; absolute pressure includes atmospheric. A tire gauge reads gauge pressure, so absolute is that plus about 101 kPa.

Why pressure is the same at equal depths

A connected fluid at rest transmits pressure through itself, so any two points at the same depth in the same fluid are at the same pressure. The basis of a manometer.

Pascal's principle

Pressure applied to an enclosed fluid is transmitted undiminished throughout. The hydraulic lift follows: a small force on a small piston balances a large force on a large one.

Hydraulic advantage

F₁/A₁ = F₂/A₂, so the force ratio equals the area ratio. The small piston travels further, so work in equals work out and nothing is created.

Buoyant force

F_b = ρ_fluid × g × V_displaced, the weight of the fluid displaced. Depends on the fluid's density and the submerged volume, not on the object's density or mass.

Archimedes' principle

The buoyant force equals the weight of the displaced fluid. Where the formula for F_b comes from.

Condition for floating

An object floats when its average density is less than the fluid's. Floating means buoyant force equals weight, so it displaces exactly its own weight of fluid.

Fraction submerged

For a floating object, submerged fraction = ρ_object / ρ_fluid. Ice at 0.92 g/cm³ in water sits about 92% under.

Apparent weight in a fluid

True weight minus the buoyant force. What a scale reads for a submerged object, and how density is measured by weighing twice.

Why a steel ship floats

Average density including the enclosed air is less than water, even though steel is denser. Shape changes the displaced volume, not the material.

Continuity equation

A₁v₁ = A₂v₂ for an incompressible fluid. Narrowing the pipe speeds the flow, because the same volume must pass every second.

Volume flow rate

Q = Av, in m³/s. Constant along a pipe with no leaks or branches.

Why faster flow means lower pressure

Bernoulli's equation: if the ½ρv² term rises at constant height, P must fall. The reason a shower curtain pulls inward.

Assumptions in Bernoulli's equation

Steady, incompressible, non-viscous flow along a streamline. Real fluids violate all three to some degree, which is why answers are approximate.

Torricelli's result

Fluid leaving a hole a depth h below the surface exits at v = √(2gh) — the same speed as an object dropped from that height.

Ideal fluid

Incompressible and non-viscous, with no energy lost to internal friction. The simplification that makes the equations in this unit solvable.

Buoyancy is a pressure difference

Pressure rises with depth, so the bottom of a submerged object is pushed up harder than the top is pushed down. Archimedes' result is that imbalance, integrated.

Buoyant force does not depend on depth

The pressure difference between top and bottom is the same at 1 m and at 100 m. What rises with depth is the absolute pressure, a different quantity.

Float, sink or hover

Compare average densities. Object less dense than fluid, it floats; denser, it sinks; equal, it is neutrally buoyant and hovers at any depth.

The floating fraction is a pure density ratio

V_sub/V_total = ρ_object/ρ_fluid. Size, shape and mass all cancel — which is why ice floats with 91.7% submerged whatever the size of the berg.

Measuring density by double weighing

Weigh an object in air, then submerged. The difference is the buoyant force, which gives the volume, which gives the density. The standard laboratory method.

Why a denser fluid means less submerged

The buoyant force must still equal the weight, and a denser fluid achieves it by displacing less volume. Swimmers float higher in the Dead Sea.

Continuity squares the radius

A₁v₁ = A₂v₂ with A = πr². Halving a pipe's radius quarters its area and QUADRUPLES the speed — not doubles it.

Bernoulli's terms are energies per unit volume

½ρv² is kinetic energy per volume, ρgh is potential energy per volume, and P is the work per volume done by the surrounding fluid. All three are in pascals.

Why a shower curtain pulls inward

Fast-moving air has a lower pressure, so the still air outside pushes the curtain in. The same effect draws two ships sailing in parallel together.

Where Bernoulli's equation fails

Viscosity in a long pipe causes a steady pressure drop it cannot predict, and turbulence destroys the streamline picture entirely. Both appear as free-response parts.

Continuity before Bernoulli

Bernoulli's equation contains two unknown speeds until continuity relates them. Reaching for Bernoulli first leaves an unsolvable equation.

Pressure is a scalar

It has no direction; the FORCE it produces acts perpendicular to whatever surface it meets. This is why pressure at a depth is the same regardless of how the surface is oriented.

A pascal is a newton per square meter

Pressure is force spread over area. The same force on half the area doubles the pressure, which is why a sharp blade cuts and a blunt one does not.

Atmospheric pressure at sea level

About 1.0 x 10^5 Pa, or 101 kPa. It is worth remembering because gauge pressure plus this number is absolute pressure.

Density of water

1000 kg/m^3, equivalently 1 g/cm^3. Most buoyancy questions are set against this value, and a fraction submerged is usually a ratio to it.

Gauge pressure can be negative

Gauge pressure is measured relative to atmospheric, so a partial vacuum reads below zero. Absolute pressure cannot be negative.

Streamline versus turbulent flow

Bernoulli's equation assumes smooth streamline flow. Once flow becomes turbulent the energy accounting stops being simple and the equation no longer applies.

Why a lower hole shoots further

Efflux speed is sqrt(2gh) with h measured from the surface down to the hole, so a deeper hole gives a faster jet. The range also depends on the fall height, so the maximum range is from the middle.

What examiners penalize here

• Distinguish gauge pressure (ρgh, the pressure from the fluid alone) from absolute pressure (P₀ + ρgh, which adds the atmosphere on top). Read the question carefully to see which is asked.
• For a floating object, the buoyant force exactly equals its weight (it is in equilibrium). The fraction submerged equals the ratio of the object’s density to the fluid’s density — a handy shortcut for iceberg-style problems.
• Continuity assumes an incompressible fluid and steady flow. On the AP exam it is almost always paired with Bernoulli’s equation: first use continuity to get the speeds, then feed them into Bernoulli for the pressures.
• Solve fluid-flow free-response in two steps: continuity (A₁v₁ = A₂v₂) gives the speeds, then Bernoulli (P + ½ρv² + ρgh = constant) gives the pressures. For horizontal pipes, drop the ρgh terms to simplify.
• On buoyancy free-responses, start by drawing a free-body diagram with weight down and buoyant force up, plus tension or a normal force if present. Every buoyancy question in this unit is a force balance, and the diagram earns its own rubric point.
• Check that every term in a Bernoulli equation comes out in pascals before you solve. P, ½ρv² and ρgh must all have the same units, and a term that does not is the fastest way to catch a substitution error under time pressure.

Practice Physics 1

Our practice bank is drawn from across the whole course rather than filtered to one unit, which is closer to how the exam asks anyway — it will not tell you which unit a question is testing.

Questions about this unit

All 8 units of AP Physics 1: Algebra-Based

1. Unit 1 · Kinematics
2. Unit 2 · Force and Translational Dynamics
3. Unit 3 · Work, Energy, and Power
4. Unit 4 · Linear Momentum
5. Unit 5 · Torque and Rotational Dynamics
6. Unit 6 · Energy and Momentum of Rotating Systems
7. Unit 7 · Oscillations
8. Unit 8 · Fluids