dimensional_analysis_1

dimensional_analysis_1 - v = C ( ) a ( p ) b , with C a...

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Dimensional Analysis Example Here is a procedure for doing systematic dimensional analysis (on the left) with an example (on the right). In the example we are looking for the dependence on environmental variables of the speed of sound v in air (or any gas). Procedure Example 1. Determine the relevant quantities from physics considerations. This may mean identifying the equation(s) that determine how the system behaves (e.g., F = ma ). The wave speed v should depend on the ambient density ρ 0 (cf. mass of particle) and the ambient pressure p 0 (cf. restoring force on particle). 2. Determine the fundamental units ([ M ], [ L ], [ T ]) of each quantity. You can use an equation that contains the quantity (e.g., F = ma ) if you know the units of everything else in the equation. v [ L ][ T ] - 1 ρ 0 mass/volume [ M ][ L ] - 3 p 0 force/area ([ M ][ L ][ T ] - 2 ) / [ L ] 2 [ M ][ L ] - 1 [ T ] - 2 3. Postulate an equation relating the quantities, with unknown exponents ( a , b , c , . . . ), which may be fractions.
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Unformatted text preview: v = C ( ) a ( p ) b , with C a dimensionless constant. 4. Substitute units from 2 and combine exponents. [ L ][ T ]-1 [ M ] a [ L ]-3 a [ M ] b [ L ]-b [ T ]-2 b = [ L ][ T ]-1 [ M ] a + b [ L ]-3 a-b [ T ]-2 b 5. Equate exponents of [ M ], [ L ], [ T ] on left and right sides of 4 and solve the resulting equations simultaneously. [ M ]: 0 = a + b [ L ]: 1 =-3 a-b [ T ]:-1 =-2 b = b = 1 / 2 = a =-1 / 2 = answer: v = C q p / 6. Always check your results by plugging back into the equations. 0 =-1 / 2 + 1 / 2 1 =-3 (-1 / 2)-1 / 2 = 3 / 2-1 / 2 -1 =-2 1 / 2 If you have identied the most relevant quantities, the (undetermined!) dimensionless coecient will typically be of order unity (e.g., between 1/3 and 3). C 1 . 2 for air!...
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