Fluids and Solids

Fluids

  • Liquids and gases are fluids.
  • Density, specific gravity
    • Density: ρ=m/V, where ρ is density, m is mass, and V is volume.
    • The density of water is ρwater = 1 g/mL = 1 g/cm3 = 1 kg/L.
    • Specific gravity is the density of something compared to water.
    • Specific gravity = ρ/ρwater.
    • The specific gravity of water is 1.
  • Buoyancy, Archimedes' principle
    • buoyancy
    • Archimedes' principle: buoyant force on an object = weight of the fluid displaced by the object.
    • FB = weightdisplaced = mdisplacedg =ρfluidVsubmergedg
    • The volume of an object that is submerged = the volume of fluid displaced by the object.
    • Things float when FB = Weight.
    • Things will rise upward when FB > Weight.
    • Things will sink when FB < Weight.
  • Hydrostatic pressure
    • Pascal's law: if you apply pressure on a liquid, the pressure is transmitted equally to all parts of the liquid.
      • pascal's law
      • F1/A1 =F2/A2
      • The pressure input at one end is the same as the pressure output at the other.
      • You apply a small force over a small area, and the output force at the end with the larger area will be greater.
      • A1d1=A2d2, where d is the distance that the end moves.
      • The work done on one end is the same as the work output at the other.
    • P = pgh (pressure vs. depth)
      • P=ρgh
      • P is pressure, ρ is the density of the fluid; g is the gravitational constant, h is the height from the surface, or depth that the object is submerged.
      • Pressure at the surface is 0 because h = 0.
      • Pressure at a depth of h is ρgh.
      • ρgh is the gauge pressure because it ignores the atmospheric pressure above the fluid.
      • Absolute pressure of something submerged in the ocean = ρgh + atmospheric pressure.
  • Viscosity: Poiseuille flow
    • When a viscous fluid flows through a pipe, the flow has a front that is shaped like a parabola bulging outward.
  • Continuity equation (A·v = constant)
    • The volume flow rate of a fluid is constant.
    • dV/dt = constant, where dV/dt is volume flow rate.
    • dV = A·dL
    • A·dL/dt = A·v = constant, where v is linear flow rate (velocity).
  • Concept of turbulence at high velocities
    • Low velocity -> laminar flow.
    • High velocity -> turbulent flow, forms eddies.
  • Surface tension
    • Surface tension gives the surface of a liquid the ability to support things that are very light.
    • For example, insects can walk on water due to surface tension.
    • Surface tension is due to the attraction between the molecules of the solvent.
  • Bernoulli's equation
    • P + ½ρv2 + ρgh = constant

Solids

  • Density: ρ=m/V, where m is mass and V is volume.
  • Elastic properties (elementary properties)
    • stress and strain
    • Stress: the pressure exerted on an object. σ = stress = F/A.
    • Strain: the deformation of the object in the direction of the applied force divided by the original length. ε = strain = ΔL/L0.
    • Young's modulus = stress / strain.
    • Young's modulus, the ratio between stress and strain, is constant until you reach the elastic limit, where things get permanently deformed.
  • Elastic limit: The maximum stress something can handle before it breaks or become permanently deformed.
  • Thermal expansion coefficient
    • Things expand when temperature rises, and contract when temperature falls.
    • ΔL = αL0ΔT
    • ΔL is the change in length, L0 is the initial length, ΔT is the change in temperature, and α is the coefficient of linear expansion.
    • In the same fashion as linear expansion, the equations for volume and area expansions are below.
    • ΔV = βV0ΔT
    • ΔA = γA0ΔT
  • Shear
    • shear
    • Shear = stress / shear ratio.
    • Shear ratio = ΔL/L0.
    • When ΔL is very small compared to L0, Shear ratio is approximately the same as the shear angle.
    • Shear angle = tan-1ΔL/L.
    • Note: ΔL and L are perpendicular to each other.
  • Compression: solids and liquids are generally not compressible. Gasses are compressible.