Formula Sheet: Fluids at Rest

Table of Contents
1. Density and Specific Gravity
2. Pressure
3. Pascal's Principle
4. Buoyancy and Archimedes' Principle
5. Surface Tension and Capillary Action
6. Pressure Measurement Devices
7. Additional Fluid Properties and Concepts
8. Important Relationships and Problem-Solving Notes
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Density and Specific Gravity

Density

• Density formula: \[\rho = \frac{m}{V}\]
  • \(\rho\) = density (kg/m³ or g/cm³)
  • \(m\) = mass (kg or g)
  • \(V\) = volume (m³ or cm³)
• Standard values:
  • Water: \(\rho_{water} = 1000\) kg/m³ = 1 g/cm³
  • Air: \(\rho_{air} \approx 1.2\) kg/m³
  • Mercury: \(\rho_{Hg} = 13,600\) kg/m³

Specific Gravity

• Specific gravity (relative density): \[SG = \frac{\rho_{substance}}{\rho_{water}}\]
  • SG = specific gravity (dimensionless)
  • \(\rho_{substance}\) = density of the substance
  • \(\rho_{water}\) = density of water (1000 kg/m³ or 1 g/cm³)
  • Note: Specific gravity has no units

Pressure

Basic Pressure

• Pressure definition: \[P = \frac{F}{A}\]
  • \(P\) = pressure (Pa, N/m², or atm)
  • \(F\) = force perpendicular to surface (N)
  • \(A\) = area (m²)
  • Note: Pressure is a scalar quantity
• Pressure units:
  • 1 atm = 101,325 Pa = 101.325 kPa
  • 1 atm = 760 mmHg = 760 torr
  • 1 Pa = 1 N/m²
  • 1 bar = 10⁵ Pa

Hydrostatic Pressure

• Absolute pressure at depth: \[P = P_0 + \rho gh\]
  • \(P\) = absolute pressure at depth (Pa)
  • \(P_0\) = atmospheric pressure at surface (typically 101,325 Pa)
  • \(\rho\) = fluid density (kg/m³)
  • \(g\) = gravitational acceleration (9.8 m/s² or 10 m/s²)
  • \(h\) = depth below surface (m)
  • Note: This assumes incompressible fluid and uniform density
• Gauge pressure: \[P_{gauge} = \rho gh\]
  • \(P_{gauge}\) = gauge pressure (pressure above atmospheric)
  • Gauge pressure does not include atmospheric pressure
  • \(P_{absolute} = P_{gauge} + P_{atm}\)
• Pressure difference between two points: \[\Delta P = \rho g \Delta h\]
  • \(\Delta P\) = pressure difference
  • \(\Delta h\) = vertical height difference
  • Note: Pressure increases with depth

Pascal's Principle

Pascal's Law

• Pascal's Principle: A change in pressure applied to an enclosed fluid is transmitted undiminished to every point in the fluid and to the walls of the container
  • Pressure changes are transmitted equally throughout the fluid
  • Basis for hydraulic systems
• Hydraulic lift/press: \[\frac{F_1}{A_1} = \frac{F_2}{A_2}\]
  • \(F_1\) = force applied to small piston (N)
  • \(A_1\) = area of small piston (m²)
  • \(F_2\) = force on large piston (N)
  • \(A_2\) = area of large piston (m²)
  • Note: Pressures are equal at same depth
• Mechanical advantage: \[MA = \frac{F_2}{F_1} = \frac{A_2}{A_1}\]
  • MA = mechanical advantage (dimensionless)
  • Larger area produces larger force
• Work conservation (ideal system): \[F_1 d_1 = F_2 d_2\]
  • \(d_1\) = distance moved by small piston
  • \(d_2\) = distance moved by large piston
  • Volume displaced is equal: \(A_1 d_1 = A_2 d_2\)

Buoyancy and Archimedes' Principle

Buoyant Force

• Archimedes' Principle: \[F_B = \rho_{fluid} V_{displaced} g\]
  • \(F_B\) = buoyant force (N, upward)
  • \(\rho_{fluid}\) = density of fluid (kg/m³)
  • \(V_{displaced}\) = volume of fluid displaced (m³)
  • \(g\) = gravitational acceleration (9.8 m/s²)
  • Note: Buoyant force equals the weight of displaced fluid
• Alternative form: \[F_B = m_{fluid} g\]
  • \(m_{fluid}\) = mass of displaced fluid

Floating and Sinking Conditions

• Object floating at equilibrium: \[F_B = W_{object}\] \[\rho_{fluid} V_{submerged} g = \rho_{object} V_{object} g\]
  • \(W_{object}\) = weight of object
  • \(V_{submerged}\) = volume of object below fluid surface
  • \(V_{object}\) = total volume of object
  • Note: Net force is zero when floating
• Fraction submerged: \[\frac{V_{submerged}}{V_{object}} = \frac{\rho_{object}}{\rho_{fluid}}\]
  • Applies to objects floating in equilibrium
  • If \(\rho_{object} < \rho_{fluid}\),="" object="">
  • If \(\rho_{object} = \rho_{fluid}\), object is neutrally buoyant
  • If \(\rho_{object} > \rho_{fluid}\), object sinks
• Apparent weight in fluid: \[W_{apparent} = W_{actual} - F_B\]
  • \(W_{apparent}\) = weight measured in fluid (scale reading)
  • \(W_{actual}\) = true weight in air
  • \(F_B\) = buoyant force

Density Determination Using Buoyancy

• Density of object from weighing: \[\rho_{object} = \frac{W_{air}}{W_{air} - W_{fluid}} \times \rho_{fluid}\]
  • \(W_{air}\) = weight in air
  • \(W_{fluid}\) = apparent weight in fluid
  • Assumes known fluid density

Surface Tension and Capillary Action

Surface Tension

• Surface tension definition: \[\gamma = \frac{F}{L}\]
  • \(\gamma\) = surface tension (N/m or J/m²)
  • \(F\) = force parallel to surface (N)
  • \(L\) = length over which force acts (m)
  • Note: Surface tension is energy per unit area
• Energy interpretation: \[\gamma = \frac{E}{A}\]
  • \(E\) = surface energy (J)
  • \(A\) = surface area (m²)
• Typical value:
  • Water at 20°C: \(\gamma \approx 0.073\) N/m

Capillary Action

• Capillary rise (or depression): \[h = \frac{2\gamma \cos\theta}{\rho g r}\]
  • \(h\) = height of liquid column rise (m)
  • \(\gamma\) = surface tension (N/m)
  • \(\theta\) = contact angle between liquid and tube
  • \(\rho\) = liquid density (kg/m³)
  • \(g\) = gravitational acceleration (m/s²)
  • \(r\) = radius of capillary tube (m)
  • Note: \(h > 0\) for wetting fluids (\(\theta < 90°\)),="" \(h="">< 0\)="" for="" non-wetting="" fluids="" (\(\theta=""> 90°\))
• Contact angle conditions:
  • Water in glass: \(\theta \approx 0°\), \(\cos\theta \approx 1\) (liquid rises)
  • Mercury in glass: \(\theta > 90°\), \(\cos\theta < 0\)="" (liquid="">

Pressure Measurement Devices

Barometer

• Atmospheric pressure from barometer: \[P_{atm} = \rho_{Hg} g h\]
  • \(P_{atm}\) = atmospheric pressure
  • \(\rho_{Hg}\) = density of mercury (13,600 kg/m³)
  • \(h\) = height of mercury column (m)
  • Standard atmosphere: \(h = 0.760\) m = 760 mmHg

Manometer

• Open-tube manometer: \[P_{gas} = P_{atm} + \rho g h\]
  • \(P_{gas}\) = pressure of gas being measured
  • \(h\) = height difference between fluid columns
  • If gas side is higher: \(P_{gas} = P_{atm} - \rho g h\)
• Closed-tube manometer: \[P_{gas} = \rho g h\]
  • One end is sealed at vacuum
  • Measures absolute pressure directly

Additional Fluid Properties and Concepts

Pressure in Multiple Fluids

• Layered fluids: \[P = P_0 + \rho_1 g h_1 + \rho_2 g h_2 + ...\]
  • Sum contributions from each layer
  • Each layer contributes \(\rho_i g h_i\) to total pressure

Compressibility and Bulk Modulus

• Bulk modulus: \[B = -V \frac{\Delta P}{\Delta V}\]
  • \(B\) = bulk modulus (Pa)
  • \(V\) = original volume
  • \(\Delta P\) = pressure change
  • \(\Delta V\) = volume change
  • Note: Negative sign ensures \(B > 0\) (volume decreases with increased pressure)
  • Higher \(B\) means less compressible (liquids have higher \(B\) than gases)
• Compressibility: \[k = \frac{1}{B}\]
  • \(k\) = compressibility (Pa^-1)
  • Inverse of bulk modulus

Cohesion and Adhesion

• Cohesion: Intermolecular forces between like molecules (within same substance)
  • Responsible for surface tension
  • Causes liquid droplets to form spheres (minimize surface area)
• Adhesion: Intermolecular forces between unlike molecules (between different substances)
  • Causes liquids to wet surfaces
  • Important in capillary action
• Meniscus formation:
  • Concave meniscus: adhesion > cohesion (e.g., water in glass)
  • Convex meniscus: cohesion > adhesion (e.g., mercury in glass)

Important Relationships and Problem-Solving Notes

Key Conceptual Points

• Pressure acts perpendicular to surfaces: Pressure at any point in a static fluid acts equally in all directions
• Pressure is independent of container shape: At a given depth, pressure depends only on depth, fluid density, and atmospheric pressure, not on container geometry
• Horizontal pressure equality: Pressure is the same at all points at the same depth in a connected fluid
• Gauge vs. absolute pressure: Always identify which type is being measured or requested
  • Absolute = Gauge + Atmospheric
• Buoyancy acts at center of buoyancy: The buoyant force acts upward through the center of mass of the displaced fluid

Common Problem Types

• Pressure at depth problems: Use \(P = P_0 + \rho gh\)
• Floating object problems: Set buoyant force equal to weight, use fraction submerged formula
• Hydraulic system problems: Use Pascal's principle with equal pressures or force ratios
• Density determination: Use buoyancy measurements and apparent weight
• Multiple fluid problems: Add pressure contributions from each layer separately

Unit Conversions (Quick Reference)

• 1 m³ = 10⁶ cm³ = 1000 L
• 1 g/cm³ = 1000 kg/m³
• 1 atm = 1.013 × 10⁵ Pa
• 1 mmHg = 133.3 Pa
• \(g\) = 9.8 m/s² ≈ 10 m/s² (approximation often used on MCAT)