1. Dimensional Reasoning

2. How many gallons are in Lake Tahoe?

3. Dimensional Reasoning
Measurements are meaningless without the correct use of units
Example: “the distance from my house to school is two”
Dimension: abstract quality of measurement without scale (i.e. length, time, mass)
Can understand the physics of a problem by analyzing dimensions
Unit: quality of a number which specifies a previously agreed upon scale (i.e. meters, seconds, grams)
SI and English units

4. Primitives
Almost all units can be decomposed into 3 fundamental dimensions (examples of units are in SI units):
Mass: M i.e. kilogram or kg
Length: L i.e. meter or m
Time: T i.e. second or s
We also have:
Luminosity i.e. candela or cd
Electrical current i.e. Ampere or A
Amount of material i.e. mole or mol

5. Derived Units (partial list)
Force
newton N LM/T2 mkg/s2
Energy
joule J L2M/T2 m2kg/s2
Pressure
pascal Pa M/LT2 kg/(ms2)
Power
watt W L2M/T3 m2kg/s3
Velocity
L/T m/s
Acceleration
L/T2 m/s2

6. Dimensional Analysis
All terms in an equation must reduce to identical primitive dimensions
Dimensions can be algebraically manipulated
examples:
Used to check consistency of equations
Can determine the dimensions of coefficients using dimensional analysis
Three equations that describe transport of “stuff”
Transport of momentum
Transport of heat
Transport of material

7. Converting Dimensions
Conversions between measurement systems can be accommodated through relationships between units
Example 1: convert 3m to cm
Example 2: 95mph fastball; how fast is this in m/s ?
1 mile = cm

8. Converting Dimensions
Conversions between measurement systems can be accommodated through relationships between units
Example 1: convert 3m to cm
Example 2: 95mph fastball; how fast is this in m/s ?
Example 3: One light-year is the distance that light travels in exactly one year. If the speed of light is 6.7 x 108 mph, convert light-years to:
a. miles
b. meters
1 mi = cm

9. Converting Dimensions
Conversions between measurement systems can be accommodated through relationships between units
Example 1: convert 3m to cm
Example 2: 95mph fastball; how fast is this in m/s ?
Example 3: One light-year is the distance that light travels in exactly one year. If the speed of light is 6.7 x 108 mph, convert light-years to:
a. miles
b. meters
Arithmetic manipulations can take place only with identical units
Example: 3m + 2cm = ?

10. Deduce Expressions for Physical Phenomena
Example: What is the period of oscillation for a pendulum?

11. Dimensionless Quantities
Dimensional quantities can be made “dimensionless” by “normalizing” with respect to another dimensional quantity of the same dimensionality
Percentages are non-dimensional numbers
Example:
Strain
Mach number
Coefficient of restitution
Reynold’s number

12. Scaling and Modeling Test large objects by building smaller models
Movies: models with scaled dimensions and scaled dynamics
Fluid dynamics: rather than studying an infinite number of pipes, understand one size very well and everything follows
Aeronautics/automotive industry: can test properties of full sized cars by building exact scaled models

13. Exothermic reaction problem. What’s the biggest elephant?
Scaling
Exothermic reaction problem. What’s the biggest elephant?

14. Thought Experiment
What would life be like on different planets? For example, on the moon with 1/6th the gravity.
How would people look?
How would bridges be different?
How would landscapes be different?