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Dimensional Reasoning

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)

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Dimensional Reasoning

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  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/T2mkg/s2 • Energy joule J L2M/T2m2kg/s2 • Pressure pascal Pa M/LT2kg/(ms2) • Power watt W L2M/T3m2kg/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 = 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 108 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 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 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 1/6th the gravity. • How would people look? • How would bridges be different? • How would landscapes be different?

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