1. Units and Dimensionality

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

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

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 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

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 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

10. Secondary Units US Speed Acceleration Power - horsepower 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 + 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?