Units and Dimensionality

Systems of Units SI – Systeme Internationale US/British Metric system MKS system US/British Non-metric Used by US and UK

Fundamental Units SI Meter Kilogram Second US/UK Foot Pound

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

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

Secondary Units Power Speed Acceleration 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

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

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

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

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

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

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?

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?

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

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

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.

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?

Dimensional Homogeneity Every term in an equation must have the same dimensionality. Consider the equation ΔG = 4πR3 Δg/3 + 4πR2 σ 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?