see also:

• a brief summary of mathematical principles and tools
• electromagnetic radiation
• light
• optics
• wind speed and wind force
• sound
• electricity in Australia
• astronomy

• velocity
  • velocity = distance travelled / unit time
  • 100kph = 100kph x 1000m / (3600secs) = 27.8m/s
• acceleration
  • acceleration = velocity change / unit time
  • a car accelerating uniformly from 0 to 100kph in 10secs will have:
    • average acceleration = 100kph x 1000m / (3600secs x 10secs) = 2.78m/second²
• momentum
  • linear momentum = mass x velocity
  • angular momentum = the moment of inertia (I) x angular velocity (ω)
    • moment of inertia (I) = mass x radius² where radius is distance to the centre of motion
      • hence a ballerina will spin faster when limbs are close to the body as moment of inertia is reduced and thus angular velocity must increase as momentum remains constant
  • NB. momentum is conserved during collisions, explosions, and other events involving objects in motion, however, in the real world, some kinetic energy is lost to heat during a collision, especially if the collision is inelastic, and thus overall momentum will fall.
  • net system momentum is constant if the net external force (or for angular momentum, external torque) is zero
• force
  • force = mass x acceleration = change in momentum / unit time
  • 1 newton (N) = 1 kilogram X meter/second²
  • lifting a 1kg object to overcome gravity requires a force of 1kg x acceleration due to gravity = 9.8N
  • force to move a 2000kg car from 0 to 100kph in 10secs = 2000 x 2.78 = 5556N
• kinetic energy
  • energy = 1/2 x mass x velocity²
  • 1 Joule = 1 kg m²/sec²
• gravitational potential energy
  • energy = mass x gravitational constant x height
• work
  • work = force x distance = change in kinetic energy
  • 1 Joule = 1Nm
  • work done to lift a 10kg object 1m off the ground = 10kg x 9.8 x 1m = 100J
  • work done to accelerate a 2000kg car from 0 to 100kph (excl. frictional and wind resistance work) = 0.5 x 2000 x (27.8m/s)² = 27,800J
• power
  • power = work / time taken = energy used / time
  • 1W = 1 Joule/sec
  • power to accelerate a 2000kg car from 0 to 100kph in 10sec = 27,800J / 10sec = 2780W
  • note that combustion engines are only about 40% efficient so the engine would actually need to use 2780/0.4 =7kW of power
  • petrol has a energy density of 44MJ/kg thus over 10sec at 7000W requires 70000J = 70kJ and thus ~44000/70 = 1.6mL of fuel IF there were no frictional forces to overcome such as wind resistance, etc.

• see also electricity in Australia
• electrical power
  • watts = current in amps x voltage
• energy capacity of a battery
  • capacity is measured in Watt-hours (Wh) ie. how many watts of power is available over 1 hour
  • watt-hours = amp-hours x nominal voltage
  • ie. a 100Ah 12.8V lithium battery will have capacity of 100 x 12.8 = 1280Wh and thus will last 12.8 hours if using 100W of power
• Ohm's law
  • voltage difference between two points in a circuit (volts) = current (amps) x resistance (ohms)

• heat energy required to heat a substance
  • heat energy = mass x specific heat x change in temperature in deg C
  • eg. water has specific heat of 4,180J/kg°C
  • eg. cast iron and stainless steel has specific heat of ~500J/kg°C
• Stefan–Boltzmann law for rate of radiant heat loss from a surface
  • radiant heat loss in watts = ϵσA(T₁⁴−T₂⁴)
  • where,
    • ϵ is the emissivity of the surface (dimensionless, between 0 and 1)
    • σ is the Stefan-Boltzmann constant (5.67×10⁻⁸ W m⁻²K⁻⁴)
    • A is the surface area (in square meters)
    • T₁ is the absolute temperature of the radiating surface (in kelvin)
    • T₂ is the absolute temperature of the surroundings (in kelvin)
• Wien's displacement law for frequency of light emitted from a black body
  • peak frequency of light =2.82k_BT/h
  • where,
    • k_B is Boltzmann's constant (1.38×10⁻²³J/K
    • T is the absolute temperature in kelvin
    • h is Planck's constant (6.63×10⁻³⁴ J·s)

• laminar flow
  • there is minimal mixing between layers of the fluid and there is very little fluctuation of the speed of the flow
  • blood flow in arteries is generally laminar
• turbulent flow
  • there is increasing mixing of the layers of flow with eddy formation resulting in varying flow rates - the greater the variation in flow, the greater the turbulence
  • energy cascade: turbulence converts kinetic energy into heat via the creation of ever smaller eddies and eventually dissipates due to viscosity
  • turbulence is complex and varies over both spatial and time dimensions
    • direct numerical simulation (DNS) can be used to model this but it is computationally complex to calculate, instead large eddy simulation (LES) is often used instead which uses a sub-grid scale model to address the small eddies
    • the Reynolds-averaged Navier-Stokes technique is the least computationally expensive model technique but this only models the effects of eddies using turbulent viscosity, not the actual eddies
    • given the computational complexity, wind tunnels are often used to assess the effect of wind on objects such as aerodynamic vehicles, tents, etc
  • turbulent flow requires more energy and results in a greater pressure drop
  • most flows in nature are turbulent
• Reynold's number
  • a value for a given system which means turbulent flow is likely to occur if Reynold's number is high for a given system
  • Reynold's number (Re) = fluid density x velocity x characteristic length / dynamic viscosity = velocity x length / kinematic viscosity = inertial forces / viscous forces
    • for flow past a cylinder, characteristic length = cylinder diameter
    • for flow past a air foil (wing of an airplane), characteristic length = length
    • for flow through a pipe:
      • characteristic length = pipe diameter
      • turbulent flow is likely if Re > 2000-4000
      • in laminar flow, flow rate at the pipe wall is zero in “no-slip condition” and velocity increases to a maximum at the centre of the pipe and the velocity profile is parabolic
      • in turbulent flow, flow rate at the pipe wall is zero in “no-slip condition” (there is a laminar sublayer here but its thickness decreases as Re increases) however, away from the wall, flow rate is more even throughout the pipe due to the mixing of layers creating a more homeogenous time averaged flow velocity but with more random variations
      • if the pipe wall roughness thickness is less than the laminar sublayer thickness then the pipe is said to be hydraulically smooth as it will not effect the turbulent flow
• Darcy-Weisbach equation to calculate pressure drop along a pipe
  • pressure drop = length of pipe x Darcy friction factor x (density/2) x (avg velocity)²/hydraulic diameter
  • for laminar flow, Darcy friction factor = 64/Re
  • for turbulent flow, there is a complicated Colebrook equation to determine the friction factor which requires an iterative calculation hence it is generally looked up on a Moody Diagram, but simplified pressure drop is proportional to velocity squared and pipe surface roughness
• Bernoulli principle and equation
  • for incompressible, non-viscous fluids:
    • fluid flow speed²/2 +gz + static pressure/density = constant
      • g = acceleration due to gravity
      • z = the elevation of the point above a reference plane
    • when applied to determine the static pressure drop when fluid flows faster through a narrowing in a pipe (the Venturi effect), the equation becomes:
      • pressure drop = density x (velocity₂² - velocity₁²)/2
      • as flow rate is equal in each pipe section, velocity₂ x pipe area₂ = velocity₁ x pipe area₁ (see also https://en.wikipedia.org/wiki/Venturi_effect))
  • see https://en.wikipedia.org/wiki/Bernoulli%27s_principle for more detail