Fluids: Density, Pressure, Buoyancy, and Fluid Dynamics (PHYS 111 Study Notes)

Fluids in Physics

Overview

Fluids are substances that can flow and take the shape of their container, including both liquids and gases. This section covers the fundamental properties and behaviors of fluids, focusing on density, pressure, static equilibrium, buoyancy, and fluid dynamics, with applications in various scientific and engineering fields.

Density

Definition and Properties

• Density (ρ) is defined as mass per unit volume.

• Formula: $\rho = \frac{M}{V}$

• Density is a key property for characterizing fluids and is often averaged over a volume for practical calculations.

Table: Densities of Common Substances

Substance	Density (kg/m3)
Gold	19,300
Mercury	13,600
Lead	11,300
Iron	7,800
Aluminum	2,700
Water (liquid, 0°C)	1,000
Ice (solid, 0°C)	917
Air	1.29
Helium	0.179
Olive oil	920
Silver	10,500
White blood cell	1,080
Cherry wood	800
Chlorine gas	3.21

Additional info: Table values are representative and may vary slightly by source.

Special Properties of Water

• Water becomes less dense when it freezes, which is unusual among substances.

• Approximately 90% of an iceberg's mass is below the water surface due to this density difference.

• Warmer water stays at the bottom of lakes, providing a stable environment for aquatic life.

Pressure

Definition and Units

• Pressure (P) is the force exerted per unit area.

• Formula: $P = \frac{F}{A}$

• SI unit: Pascal (Pa), where $1~\text{Pa} = 1~\text{N/m}^2$.

• Atmospheric pressure at sea level: $P_{\text{at}} = 1.01 \times 10^5~\text{N/m}^2$

• Pressure differences generate net forces in fluids.

• Key Concept: You do not suck liquid through a straw; instead, atmospheric pressure pushes the liquid up when you reduce the pressure inside the straw.

Pressure and Depth

Hydrostatic Pressure

• Pressure in a fluid increases with depth due to the weight of the fluid above.

• At a depth h below the surface: $P = P_{\text{at}} + \rho g h$

• Pressure is the same at all points at the same depth in a connected fluid.

• Dams are thicker at the base to withstand higher pressures at greater depths.

Dependence of Pressure on Depth

• For two points at depths $h_1$ and $h_2$: $P_2 = P_1 + \rho g (h_2 - h_1)$

• The difference in pressure is due to the weight of the fluid column between the two points.

Applications of Pressure

Barometer

• A barometer measures atmospheric pressure using a column of fluid (often mercury).

• The top of the column is a vacuum, so the fluid rises until its weight balances the atmospheric pressure.

• This is an example of pressure equilibrium in fluids.

Pascal's Principle

• Any change in pressure applied to an enclosed fluid is transmitted undiminished throughout the fluid.

• Formula for hydraulic systems: $F_1 d_1 = F_2 d_2$

• Applications include hydraulic lifts and brakes.

Buoyancy and Archimedes' Principle

Buoyant Force

• The buoyant force on an object submerged in a fluid is equal to the weight of the fluid displaced by the object.

• Formula: $F_{\text{buoyant}} = \rho_{\text{fluid}} V_{\text{displaced}} g$

• The pressure on the sides of the object cancels out; only the difference between top and bottom pressures matters.

• The buoyant force depends on the fluid's density, not the object's density.

• If the buoyant force is greater than the object's weight, the object will float.

Floating and Sinking

• Objects can float even if made from dense materials, provided they displace enough fluid (e.g., boats made of concrete with air pockets).

• Equilibrium occurs when the buoyant force equals the object's weight.

Fluid Flow and Continuity

Conservation of Mass in Fluids

• For an incompressible fluid, the mass flow rate is constant throughout a pipe or vessel.

• Equation of Continuity: $\rho_1 A_1 v_1 = \rho_2 A_2 v_2$

• For liquids (incompressible), $\rho_1 = \rho_2$, so: $A_1 v_1 = A_2 v_2$

• This principle applies to blood flow, pipelines, and other systems involving fluid transport.

Bernoulli's Equation and Fluid Energy

Work-Energy Principle for Fluids

• The kinetic energy of a fluid element: $K = \frac{1}{2} (\Delta m) v^2 = \frac{1}{2} (\rho \Delta V) v^2$

• Bernoulli's Equation (for steady, incompressible, non-viscous flow): $P_1 + \frac{1}{2} \rho v_1^2 + \rho g y_1 = P_2 + \frac{1}{2} \rho v_2^2 + \rho g y_2$

• This equation expresses the conservation of energy for flowing fluids, relating pressure, kinetic energy per unit volume, and potential energy per unit volume.

• When velocity increases, pressure decreases (basis for lift in airplane wings).

Applications of Bernoulli's Equation

• If a hole is punched in the side of a container, the fluid at the hole's level is at higher pressure than the outside, causing fluid to exit with horizontal velocity.

• If the fluid is directed upward, it can reach the same height as the fluid surface inside the container (demonstrating conservation of energy).

Summary Table: Key Fluid Equations

Concept	Equation	Description
Density	$\rho = \frac{M}{V}$	Mass per unit volume
Pressure	$P = \frac{F}{A}$	Force per unit area
Hydrostatic Pressure	$P = P_{\text{at}} + \rho g h$	Pressure at depth h
Buoyant Force	$F_{\text{buoyant}} = \rho_{\text{fluid}} V_{\text{displaced}} g$	Upward force on submerged object
Continuity	$A_1 v_1 = A_2 v_2$	Conservation of mass for incompressible fluids
Bernoulli's Equation	$P_1 + \frac{1}{2} \rho v_1^2 + \rho g y_1 = P_2 + \frac{1}{2} \rho v_2^2 + \rho g y_2$	Conservation of energy for fluids

Applications and Examples

• Human Biology: Blood flow in arteries and veins follows the continuity equation and Bernoulli's principle.

• Meteorology: Atmospheric pressure and barometers are essential for weather prediction.

• Engineering: Hydraulic lifts use Pascal's principle; dams are designed considering hydrostatic pressure.

• Astrophysics: Fluid dynamics applies to gas clouds and stellar atmospheres.