Chapter 13: Fluid Statics (Sections 1–3) – Mass Density, Pressure, and Archimedes’ Principle

Chapter 13: Fluid Statics

Overview

This chapter introduces the fundamental concepts of fluid statics, focusing on mass density, pressure in fluids, and Archimedes’ Principle. These principles are essential for understanding how fluids behave at rest and how they exert forces on objects and surfaces.

Mass Density

Definition and Formula

• Mass density (ρ) of a material is the amount of mass per unit volume.

• If mass is uniformly distributed, the density is given by:

$\rho = \frac{m}{V}$

• Units: The SI unit of density is kilograms per cubic meter (kg/m3).

Typical Densities of Materials

The table below lists the densities of common materials:

Material	Density (kg/m3)	Material	Density (kg/m3)
Air (1 atm, 20°C)	1.20	Aluminum	2.7 × 103
Ethanol	0.806 × 103	Copper	8.9 × 103
Water	1.00 × 103	Lead	11.3 × 103
Seawater	1.03 × 103	Gold	19.3 × 103
Blood	1.06 × 103	Platinum	21.4 × 103
Styrofoam	0.05 × 103	White-dwarf star	109

Additional info: Table values inferred from the provided image and standard physics tables.

Specific Gravity (Relative Density)

• Specific gravity is the ratio of the density of a material to the density of water at 4°C.

• It is a dimensionless quantity (no units).

• Formula:

$\text{Specific gravity} = \frac{\rho_{\text{material}}}{\rho_{\text{water}}}$

• For example, $\rho_{\text{water}} = 1.00 \times 10^3$ kg/m3, $\rho_{\text{gold}} = 19.3 \times 10^3$ kg/m3, so the specific gravity of gold is 19.3.

• Specific gravity is sometimes called relative density.

Pressure in Fluids

Definition of Pressure

• Pressure (p) in a fluid is the perpendicular force (F⊥) per unit area (A) exerted on a surface.

• Formula:

$p = \frac{F_{\perp}}{A}$

• The force exerted by a fluid is always perpendicular to the surface at each point (for fluids at rest).

Pascal’s Law

• Pascal’s Law: Pressure applied to an enclosed fluid is transmitted undiminished to every portion of the fluid and the walls of the containing vessel.

• The pressure at a given depth depends only on the depth, not on the shape of the container.

Units of Pressure

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

• Other common units:

  • 1 atm = $1.01325 \times 10^5$ Pa ≈ 14.7 psi

  • 1 bar = 1000 millibars = $10^5$ Pa

  • 1 torr = $\frac{1}{760}$ atm

  • 1 mm Hg ≈ 133.3 Pa (pressure from a 1 mm column of mercury)

Pressure Variation with Depth

• The total pressure at a depth h in a fluid of density ρ is:

$p = p_0 + \rho g h$

• p0 is the pressure at the surface (often atmospheric pressure).

• h is the depth below the surface.

• This equation is used to calculate the pressure you feel underwater or at any depth in a fluid.

Gauge Pressure and Absolute Pressure

• Gauge pressure: The pressure measured relative to atmospheric pressure (e.g., tire pressure gauges).

• Absolute pressure: The total pressure, including atmospheric pressure.

• Relationship: $p_{\text{abs}} = p_{\text{gauge}} + p_{\text{atm}}$

Hydraulic Systems

• Hydraulic systems use Pascal’s law to multiply force.

• For two pistons connected by a fluid:

$p_1 = p_2 \implies \frac{F_1}{A_1} = \frac{F_2}{A_2}$

• This principle is used in car lifts and hydraulic presses.

Archimedes’ Principle

Statement of the Principle

• When an object is completely or partially immersed in a fluid, the fluid exerts an upward buoyant force equal to the weight of the fluid displaced by the object.

• Formula for buoyant force:

$F_B = m_{d}g = \rho_{d} V_{d} g$

• Where $m_{d}$ is the mass of displaced fluid, $\rho_{d}$ is the density of the displaced fluid, $V_{d}$ is the volume of displaced fluid, and $g$ is the acceleration due to gravity.

Example Problem

• A foam waterboard (5 cm × 60 cm × 120 cm, density = 40 kg/m3) is fully submerged underwater.

• a) The buoyant force acting on it is 353 N (≈ 79 lb).

• b) The force needed to keep it submerged is 339 N.

Additional info: The calculation uses the volume of the board and the density of water to find the buoyant force, then subtracts the board’s weight to find the force needed to keep it submerged.

Summary Table: Key Fluid Statics Quantities

Quantity	Symbol	Formula	SI Unit
Density	$\rho$	$\rho = \frac{m}{V}$	kg/m3
Pressure	$p$	$p = \frac{F_{\perp}}{A}$	Pa (N/m2)
Hydrostatic Pressure	$p$	$p = p_0 + \rho g h$	Pa
Buoyant Force	$F_B$	$F_B = \rho_{d} V_{d} g$	N

Applications

• Understanding why objects float or sink in fluids (boats, icebergs, balloons).

• Design of hydraulic machines and pressure measurement devices.

• Atmospheric and oceanic pressure calculations.