In-Depth Notes on Hydrostatics

Expected Learning Outcomes

• Students should be able to:
  • Explain density, specific gravity, fluid pressure, and pressure of fluid inside a vessel.
  • Compute the effect of fluid on submerged bodies.

Hydrostatic Overview

• Hydrostatics: Study of fluids at rest, typically confined within a vessel.
• Types of fluids:
  • Liquids (e.g., bottled mineral water, blood, hydraulic fluid)
  • Gases (e.g., air)

Key Properties of Materials

Density

• Defined as mass per unit volume.
• Formula: $p = \frac{m}{V}$ where:
  • $p$ = density in g/cm³ or kg/m³
  • $m$ = mass in grams or kg
  • $V$ = volume in cm³ or m³

Common Densities

• Copper: $p = 8.9 ext{ g/cm}^3$
• Lead: $p = 11.3 ext{ g/cm}^3$
• Water: $p = 1 ext{ g/cm}^3$ (or 1000 kg/m³)
• Useful conversion: 1 ml = 1 cc = 1 cm³

Specific Gravity (sp.gr)

• Defined as the ratio of the density of a substance to the density of a standard substance (usually water).
  • Formula:
    $sp.gr. = \frac{Density<em>{substance}}{Density</em>{standard}}$
• Unitless value approximately equals the density.

Example Calculations

Density and Specific Gravity of Glycerin

• Given: Mass = 60.4 g, Volume = 48 ml.
  • Density Calculation:
    $p = \frac{60.4 ext{ g}}{48 ext{ ml}} = 1.26 ext{ g/cm}^3$
  • Specific Gravity Calculation:
    $sp.gr. = \frac{1.26}{1} = 1.26$

Applications

• Blood density (normal: 1.04 to 1.06 g/cm³), indicating anemia with unusually low values.
• Urine density (normally around 1.02 g/cm³), indicating salt excretion levels.

Weight Density

• Defined as weight per unit volume.
• Formula:
  $D = \frac{Weight}{Volume}$
• Units: N/m³, dynes/cc, lbs/ft³ for water: 62.4 lbs/ft³

Hydrostatic Pressure

• The pressure exerted by a column of liquid in a vessel.
• Formula: $P = \frac{F}{A}$ where:
  • $P$ = pressure (N/m²)
  • $F$ = force (N)
  • $A$ = area (m²)

Depth and Pressure

• Pressure depends on the depth of the liquid and its weight density: $P = \rho gh$ where:
  • $\rho$ = density of liquid
  • $g$ = gravitational acceleration (9.8 m/s²)
  • $h$ = height of liquid column

Example: Hydrostatic Pressure Exerted by Mercury

• Given: Density of mercury = 13.6 g/cm³, height = 12 cm.
  • Pressure Calculation:
    $P = (13.6 ext{ g/cm}^3)(980 ext{ cm/s}^2)(12 ext{ cm}) = 159,936 ext{ dyne/cm}^2$

Pascal’s Principle

• States that pressure applied to an enclosed fluid is transmitted undiminished throughout the fluid and walls of the container.

Hydraulic Systems

Pascal’s Law for Hydraulic Systems

• Relationship between force, area, and pressure:
  • $P<em>1 = P</em>2$
  • Thus,
  • $\frac{F<em>1}{A</em>1} = \frac{F<em>2}{A</em>2}$

Example: Hydraulic Press

• Given:
  • Small piston area = 0.005 m²
  • Large piston area = 0.03 m²
  • Force applied to small piston: 500 N
  • Force on larger piston:
    $F<em>2 = F</em>1 \frac{A<em>2}{A</em>1} = 500 \frac{0.03}{0.005} = 3000 ext{ N}$

Buoyancy and Archimedes’ Principle

• States that an immersed body experiences an upward force equal to the weight of the fluid it displaces.

Buoyant Force (BF)

• Formula:
  $BF = W<em>{air} - W</em>{liquid}$

Conditions for Floating and Sinking

1. Weight of the body < Buoyant Force: Floats
2. Weight of the body > Buoyant Force: Sinks
3. Weight of the body = Buoyant Force: Suspended

Example: Solid Weighing in Different Mediums

• Given: Wair = 350 dynes, Wwater = 210 dynes.
  • Buoyant Force Calculation:
    $BF = W<em>{air} - W</em>{water} = 350 ext{ dynes} - 210 ext{ dynes} = 140 ext{ dynes}$
  • Specific Gravity Calculation:
    $sp.gr. = \frac{Weight<em>{object}}{Weight</em>{displaced}} = \frac{350}{140} = 2.5$