AP Physics: Understanding Fluids (Algebra-Based Study Notes)

Fluid

A substance that can flow and take the shape of its container; includes liquids and gases.

Liquid

A fluid with a definite volume that flows and takes the shape of its container (e.g., water).

Gas

A fluid that expands to fill its container (e.g., air).

Continuum model

Treating a fluid as a smooth, continuous material instead of tracking individual molecules; works well at everyday scales.

Density (ρ)

Mass per unit volume; ρ = m/V (SI unit: kg/m^3).

Pressure (P)

Force distributed over area; P = F/A, where F is the perpendicular (normal) force.

Pascal (Pa)

SI unit of pressure; 1 Pa = 1 N/m^2.

Normal (perpendicular) force

The component of force perpendicular to a surface; the relevant force component when computing pressure via P = F/A.

Atmospheric pressure (P_atm)

The pressure exerted by the atmosphere; often used as the reference level for gauge pressure and added for absolute pressure.

Absolute pressure (P_abs)

Pressure measured relative to a perfect vacuum; Pabs = Patm + P_gauge.

Gauge pressure (P_gauge)

Pressure measured relative to atmospheric pressure (what many everyday gauges read).

Hydrostatic equilibrium

A condition where a fluid is at rest (not moving).

Pressure-depth relationship

For a fluid at rest open to the atmosphere: P = P_atm + ρgh, where h is vertical depth below the surface.

Pressure difference with depth (ΔP)

The increase in pressure between two depths in a static fluid: ΔP = ρgh.

Same-depth pressure in a connected fluid

In a static fluid of uniform density, all points at the same depth have the same pressure (independent of container shape).

Manometer

A device that measures pressure differences using a fluid column (often in a U-tube), relying on equal pressures at the same level in connected fluid.

Open-tube manometer

A manometer with one side open to the atmosphere; relates gas pressure to atmospheric pressure by Pgas − Patm = ρgΔh.

Barometer

A device that measures atmospheric pressure by balancing it against the weight of a liquid column (uses hydrostatic pressure ideas).

Pascal’s principle

A pressure change applied to a confined fluid at rest is transmitted throughout the fluid.

Confined (enclosed) fluid

A sealed fluid system (e.g., in a hydraulic lift) where applied pressure changes transmit through the fluid.

Hydraulic press

A system using pistons and a confined fluid to multiply force using area differences (foundation of hydraulic lifts/brakes).

Force multiplication (hydraulics)

In an ideal hydraulic system, equal pressures give F1/A1 = F2/A2, so F2 = F1(A2/A1).

Volume conservation in hydraulics

For piston motion in an incompressible confined fluid: A1 d1 = A2 d2 (displaced volumes match).

Work conservation in ideal hydraulics

Ideally (no losses), input and output work match: F1 d1 = F2 d2; larger force comes with smaller distance.

Buoyant force (F_B)

Net upward force on an object in a fluid due to higher pressure at greater depth.

Archimedes’ principle

The buoyant force equals the weight of the displaced fluid: FB = ρfluid g V_disp.

Displaced volume (V_disp)

The volume of fluid displaced by an object; equals the object’s submerged volume (not necessarily its total volume).

Apparent weight (W_app)

The support force needed when buoyancy assists; for an object held submerged: Wapp = W − FB.

Neutrally buoyant

Condition where buoyant force equals weight (F_B = W), so the object can remain in equilibrium in the fluid.

Density criterion for float/sink

For a fully submerged object: it rises if ρobj < ρfluid, sinks if ρobj > ρfluid.

Submerged fraction

For a floating object at rest: Vsub/Vobj = ρobj/ρfluid.

Streamline

A path in a flowing fluid that is tangent to the velocity direction at every point.

Ideal fluid flow assumptions

Model of flow assuming incompressible, nonviscous, steady flow along streamlines (used for continuity and Bernoulli).

Incompressible fluid

A fluid whose density is effectively constant (a key assumption in ideal flow and continuity).

Nonviscous flow

Flow with negligible internal friction (viscosity), allowing mechanical energy conservation in Bernoulli’s equation.

Steady flow

Flow where properties at a point (speed, pressure, etc.) do not change with time.

Volume flow rate (Q)

Volume per time passing a point; Q = ΔV/Δt and for a pipe Q = Av.

Continuity equation

For steady incompressible flow: A1 v1 = A2 v2 (flow rate is the same along the pipe).

Venturi effect (conceptual)

In many ideal-flow situations, a narrower region increases speed and is associated with lower static pressure (using continuity + Bernoulli).

Bernoulli’s equation

Energy conservation per volume along a streamline for steady, incompressible, nonviscous flow: P + (1/2)ρv^2 + ρgy = constant.

Kinetic energy per volume term (½ρv²)

The Bernoulli term representing flow kinetic energy density; increases when speed increases.

Gravitational potential energy per volume term (ρgy)

The Bernoulli term representing gravitational potential energy density; increases with height y.

Bernoulli limitations

Bernoulli may fail or need extra terms for unsteady, compressible, highly viscous, turbulent flow, or when pumps/turbines add/remove energy.

Torricelli’s law

For outflow from a small hole at depth h in a large tank open to air: v = √(2gh) (from Bernoulli).

Depth-to-hole height (h) in Torricelli problems

In v = √(2gh), h is the vertical distance from the free surface to the hole (not necessarily total tank depth).

Viscosity

A measure of a fluid’s internal friction/resistance to flow; causes energy losses and pressure drops in real fluids.

Laminar flow

Smooth, layered flow with predictable streamlines; often at lower speeds or higher viscosity.

Turbulent flow

Chaotic flow with eddies and mixing; often at higher speeds or around obstacles, typically with significant energy losses.

Drag force

A resistive force opposite an object’s motion relative to the fluid; generally increases with speed and depends on shape/area/flow regime.

Terminal speed

The constant speed reached when drag force balances weight (net force becomes zero, so acceleration becomes zero).