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: Mi.e. kilogram or kg  Length: L i.e. meter or m  Time: Ti.e. second or s We also have:  Luminosityi.e. candela or cd  Electrical currenti.e. Ampere or A  Amount of materiali.e. mole or mol

5. Derived Units (partial list) Force newtonNLM/T 2 mkg/s 2 Energy jouleJL 2 M/T 2 m 2 kg/s 2 Pressure pascalPaM/LT 2 kg/(ms 2 ) Power watt WL 2 M/T 3 m 2 kg/s 3 Velocity L/Tm/s Acceleration L/T 2 m/s 2

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 = 160934.4 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 10 8 mph, convert light-years to: a. miles b. meters 1 mi = 160934.4 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 10 8 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 http:///www.wetanz.com/models-miniatures http://www.colorado.edu/aerospace/vs_focus.html

13. 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 1/6 th the gravity.  How would people look?  How would bridges be different?  How would landscapes be different?