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: 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 materiali.e. mole or mol

Derived Units (partial list)

Forcenewton N LM/T2 mkg/s2

Energyjoule J L2M/T2 m2kg/s2

Pressurepascal Pa M/LT2 kg/(ms2)

Powerwatt W L2M/T3 m2kg/s3

VelocityL/T m/s

AccelerationL/T2 m/s2

Dimensional Analysis

All terms in an equation must reduce to identical primitive dimensions

Dimensions can be algebraically manipulatedexamples:

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 = 160934.4 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 108 mph, convert light-years to:

a. milesb. meters

1 mi = 160934.4 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 108 mph, convert light-years to:

a. milesb. 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

http:///www.wetanz.com/models-miniatureshttp://www.colorado.edu/aerospace/vs_focus.html

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 1/6th the gravity.

How would people look? How would bridges be different? How would landscapes be different?