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Units and Dimensionality

Units and Dimensionality. Systems of Units. SI – Systeme Internationale Metric system MKS system US/British Non-metric Used by US and UK. Fundamental Units. SI Meter Kilogram Second US/UK Foot Pound Second. SI System. Primary / Fundamental units Unit of length - Meter - m

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Units and Dimensionality

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  1. Units and Dimensionality

  2. Systems of Units • SI – SystemeInternationale • Metric system • MKS system • US/British • Non-metric • Used by US and UK

  3. Fundamental Units • SI • Meter • Kilogram • Second • US/UK • Foot • Pound • Second

  4. SI System • Primary / Fundamental units • Unit of length - Meter - m • Unit of mass – kilogram – kg • Unit of time – second – s • Unit of temperature – Kelvin – K • Unit of charge – Coulomb – C

  5. SI Secondary/Derived Units • Unit of force – Newton – N • F = ma • 1 Newton = 1 kg x 1 m/s2 • Unit of Work/Energy • W = F x displacement • 1 Joule = 1 Newton x 1 meter = 1 N-m • Energy is also expressed in Joules

  6. Secondary Units • Power • P = W/t • 1 Watt = 1 Joule/sec = 1 J/s = 1 N-m/s • Speed • v = dx/dt = distance/time • v = meters/sec = m/s • Acceleration • a = dv/dt = meters/sec/sec = m/s2

  7. Secondary Units • Current • I = dQ/dt • 1 Ampere = 1 Coulomb/sec • 1 A = 1C/s

  8. US System • Primary units • Foot - length - ft • Force – pound – lb • Time – second – s Some people refer to the unit of force in the US system as a pound force or lbf

  9. Secondary Units - US • Mass – slug • F = ma • M = F/a • 1 slug = 1 pound/(1 ft/s2) • Work • W = Force x displacement • 1 ftlb = 1 lb x 1 ft • Energy is expressed in ftlb as well

  10. Secondary Units US • Speed • v = dx/dt • v – ft/s • Acceleration • a = dv/dt • a = ft/s2 • Power - horsepower

  11. SI - Prefixes • Giga x 109 • Mega x 106 • Kilo x 103 • Deci x 10-1 • Centi x 10-2 • Milli x 10-3 • Micro x 10-6 • Nano x 10-9 • Pico x 10-12

  12. Class Exercise 1 • A car is traveling at a speed of 65 mph on I195E. • What is its speed in ft/s? • What is its speed in kmh? • What is its speed in m/s?

  13. Class Exercise 2 • What is your own height in feet and inches? • Convert your height to meters. • What is your own weight in pounds? • Is weight a force or a mass? • Convert your weight to oz? • Convert your weight to Newtons. • What is your mass in the Si system? In the US system?

  14. Dimensionality • There is a difference between units and dimensions/dimensionality. • Distance is expressed in units of length, e.g. m or ft. • The dimensionality of distance if L • The dimensionality of mass is M • The dimensionality of time is T

  15. Dimensionality (cont’d) • Force has a dimensionality of ? • F = m x a • [F] means the dimensionality of force • [F] = [m] x [a] = M x L/T2 • W = F x distance • [W] = [F] x [distance] = (M x L/T2) x L • [W] = ML2 /T2

  16. Special Cases • The argument of a trigonometric, logarithmic, or exponential function , e.g. sin (x), log (x), exp (x) must be dimensionless. • In the function sin ϴ, the argument ϴ must be dimensionless. • Degrees or radians are dimensionless • In the function exp (-Q/kT), the argument must be dimensionless.

  17. Class Exercise 3 • Consider the function exp (-Q/kT), where Q is an energy and T is a temperature. k is the Boltzmann constant. • Determine the dimensionality of k. • What would be the units of k in the SI system?

  18. Dimensional Homogeneity • Every term in an equation must have the same dimensionality. • Consider the equation ΔG = 4πR3 Δg/3 + 4πR2σ Where ΔG is an energy, R is the radius of a sphere, Δg is an energy per unit volume, and σ is a surface energy. • Is this equation dimensionally homogeneous?

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