Unit 8 Beta

Fluids: the one-page sheet

8.1 Internal Structure and Density

Fluids (liquids and gases) flow because their particles can move past one another; density ρ = m/V is a property of the material, and an ideal fluid is incompressible and nonviscous, so its density is constant.

A fluid is a substance that flows: a liquid or a gas. Its particles can slide past each other, so it has no fixed shape.
Density is mass per volume, ρ = m/V, in kg/m³. It belongs to the material: cut a sample in half and its density does not change.
Unit check: 1 g/cm³ = 1000 kg/m³. Water is about 1000 kg/m³, aluminum 2700 kg/m³, air about 1.2 kg/m³.
An ideal fluid is incompressible (constant density) and nonviscous (no internal friction). Treat every fluid on the exam as ideal unless told otherwise.

the solid keeps its own shape the material flows and takes its container's shape: it is a fluid a liquid is nearly incompressible, so its density stays almost constant a fluid is described by one fixed density ρ = m/V

state of matter: One of the forms matter takes (solid, liquid or gas), set by how closely its particles are packed and how freely they move.
fluid: A substance that flows and takes the shape of its container, because its particles can move past one another: a liquid or a gas.
density: Mass per unit volume, ρ = m/V, in kg/m³. A property of the material, not of the amount; water is about 1000 kg/m³.
ideal fluid: A model fluid that is incompressible (constant density) and nonviscous (no internal friction), flowing smoothly; the exam's fluids are ideal unless stated.

8.2 Pressure

Pressure is perpendicular force per area, a scalar in pascals; in a fluid at rest the absolute pressure is P = P₀ + ρgh, set only by depth, and the gauge pressure (what gauges read) is ρgh.

Pressure is perpendicular force per area, P = F⊥/A, in pascals (1 Pa = 1 N/m²). It is a scalar: the force it makes on any surface points straight into that surface.
In a fluid at rest, P = P₀ + ρgh. Pressure depends only on depth, not on the container's shape or the amount of fluid.
Absolute pressure P includes the atmosphere's P₀ ≈ 1.0 × 10⁵ Pa. Gauge pressure P − P₀ = ρgh is what a gauge or tire gauge reads.
About 10 m of water adds one atmosphere of pressure.

they push on it, perpendicular to the surface, with a pressure P = F⊥/A pressure increases with depth the pressure grows by ρgh: P = P₀ + ρgh it measures gauge pressure, ρgh, not the absolute pressure P₀ + ρgh

pressure: The perpendicular force per unit area on a surface, P = F⊥/A, a scalar measured in pascals (1 Pa = 1 N/m²).
hydrostatic pressure: The pressure at a depth h in a fluid at rest: P = P₀ + ρgh, set by depth, not by the container's shape.
gauge pressure: The pressure above atmospheric pressure, P_gauge = P − P₀ = ρgh in a liquid; absolute pressure P includes the atmosphere's P₀ ≈ 1.0 × 10⁵ Pa.

8.3 Fluids and Newton's Laws

The buoyant force is the fluid's net upward push, caused by pressure increasing with depth; it equals the weight of the displaced fluid, ρ_fluid V_displaced g, and Newton's laws with buoyancy and gravity as separate forces decide whether an object floats, sinks or accelerates.

The buoyant force is the fluid's net upward push. On a free-body diagram it is a separate arrow, up, labeled "F_b, water on block", opposite to the gravitational force.
Archimedes' principle: F_b = ρ_fluidV_displacedg, the weight of the displaced fluid. It depends on the fluid's density and the displaced volume, not on the object's density or weight.
A floating object is in equilibrium: F_b = mg, so the fraction under the surface is ρ_object/ρ_fluid.
A submerged object denser than the fluid sinks with a = g(1 − ρ_fluid/ρ_object), neglecting drag.

the fluid pushes up on an object's bottom harder than it pushes down on its top the net push of the fluid is upward: the buoyant force the buoyant force is ρ_fluid V_displaced g, the weight of the displaced fluid it floats where F_b = mg, sinks if mg is larger, and accelerates at a = (mg − F_b)/m

fluid parcel: A small, imagined portion of a fluid treated as an object, so Newton's laws can be applied to it.
buoyant force: The net upward force a fluid exerts on an object in it, because the pressure on the object's bottom is larger than on its top.
Archimedes' principle: The buoyant force equals the weight of the fluid the object displaces: F_b = ρ_fluid V_displaced g. It does not depend on the object's own density.
neutral buoyancy: The condition in which the buoyant force on an object equals its weight, so it neither rises nor sinks. A fully submerged object is neutrally buoyant when its average density equals the fluid's; a floating object reaches the same balance with only part of it under the surface.

8.4 Fluids and Conservation Laws

For an ideal fluid in a full pipe, continuity (A₁v₁ = A₂v₂, mass conservation) sets the speeds and Bernoulli's equation (energy conservation per volume) then sets the pressures; faster flow at the same height means lower pressure, and a hole a depth h down gives v = √(2gh).

Volume flow rate is Av, in m³/s. For an ideal fluid in a full pipe it is the same everywhere: A₁v₁ = A₂v₂ (continuity, from conservation of mass).
Continuity sets the speed: a quarter of the area gives four times the speed. Use it first.
Bernoulli: P₁ + ρgy₁ + ½ρv₁² = P₂ + ρgy₂ + ½ρv₂², conservation of energy per volume. At equal heights, faster flow means lower pressure.
Torricelli: fluid leaves a small hole a depth h below an open surface at v = √(2gh), the speed of a drop falling h.

the volume entering each second equals the volume leaving: A₁v₁ = A₂v₂ the fluid must move faster there (continuity sets the speed) the pressure is lower where the fluid moves faster, at the same height P + ρgy + ½ρv² is the same at every point along the flow (Bernoulli)

volume flow rate: The volume of fluid passing a point each second, Av (area × speed), in m³/s.
continuity equation: For an incompressible fluid in a full pipe, the volume flow rate is the same everywhere: A₁v₁ = A₂v₂. It expresses conservation of mass.
Bernoulli's equation: P₁ + ρgy₁ + ½ρv₁² = P₂ + ρgy₂ + ½ρv₂² along an ideal fluid's flow: conservation of energy per unit volume.
Torricelli's theorem: The speed of fluid leaving a small opening a depth h below an open surface: v = √(2gh), derived from Bernoulli's equation.