Dimensional Reasoning

How many gallons are in Lake Tahoe?

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

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

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

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

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

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 = cm

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 = ?

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

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

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

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

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?