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Example Solutions Homework 11

In the case above length time and mass appear in

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Unformatted text preview: above, Length, time, and Mass appear in multiple variables; therefore there are 3 reductions. 4) The number of repeating variables is equal to the number of reductions, 3. The repeating variables must be the least complicated in terms of their dimensions while making sure that every dimension is represented. Therefore, choose d, t, and . 5) Generate terms equal to the number of variables (7) minus the number of reductions (3) = 4 terms Generate 1: (a) Multiply the dependent variable by the repeating variables: Π ∆h d t ρ (b) Set the left hand side exponents equal to the right hand exponents (dimensionally): M L t L L t M L (c) Solve for the exponents by forcing x,y, and z to be zero, in turn: M: 0 = (1) x c1 c1 = 0 t: 0 = (1) x b1 b1 = 0 L: 0 = 1 + (1) x a1 + (-3) x c1 a1 = -1 + (3)c1 a1 = -1 (d) Substitute the exponents back into the original equation: Π ∆h d t ρ Therefore, Π ∆h d Generate 2: (a) Multiply one of the remaining independent variables by the repeating variables: Π D d t ρ (b) Set the left hand side exponents equal to the right hand exponents (dimensionally): M L t L L t M L (c) Solve for the exponents by forcing x,y, and z to be zero, in turn: M: 0 = (1) x c2 c2 = 0 t: 0 = (1) x b...
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