p0294

# p0294 - 2.94 CHAPTER 2 PROBLEM 94 241 2.94 Chapter 2...

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2.94. CHAPTER 2, PROBLEM 94 241 2.94 Chapter 2, Problem 94 Figure 2.24: Chihuahua with powerful rear-leg muscles. 2.94(a): The dimensional quantities and their dimensions are [ E ] = F · L = ML/T 2 · L = ML 2 T 2 [ H ] = L, [ M ] = M, [ g ] = L T 2 There are 4 dimensional quantities and 3 independent dimensions ( M, L, T ) , so that the number of dimensionless groupings is 1. Using the indicial method, the appropriate dimensional equation is [ H ] = [ M ] a 1 [ g ] a 2 [ E ] a 3 Substituting the dimensions for each quantity yields L = M a 1 L a 2 T 2 a 2 M a 3 L 2 a 3 T 2 a 3 = M a 1 + a 3 L a 2 +2 a 3 T 2 a 2 2 a 3 Thus, equating exponents, we arrive at the following three equations: 0 = a 1 + a 3 1 = a 2 + 2 a 3 0 = 2 a 2 2 a 3 We can solve immediately for a 2 and a 3 from the first and third equations, viz., a 1 = a 3 and a 2 = a 3

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242 CHAPTER 2. DIMENSIONAL ANALYSIS Substituting into the second equation yields 1 = a 3 + 2 a 3 = a 3 = a 3 = 1 Consequently, a 1 = 1 and a 2 = 1 Substituting back into the dimensional equation, we have [ H ] = [ M ] 1 [ g ] 1 [ E ] Therefore, the dimensionless grouping is MgH E Using E. S. Taylor’s method, we have the following. M L T H 0 1 0 M 1 0 0 g 0 1 2 E 1 2 2 M L T H 0 1 0 M/E 0 2 2 g 0 1 2 E 1 2 2 L T H 1 0 M/E 2 2 g 1 2
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