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p0238 - 2.38. CHAPTER 2, PROBLEM 38 153 2.38 Chapter 2,...

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2.38. CHAPTER 2, PROBLEM 38 153 2.38 Chapter 2, Problem 38 Figure 2.5: Capillary tube in a pan of liquid. The dimensional quantities and their dimensions are [ σ ]= F L = ML/T 2 L = M T 2 [ h L, [ ρ M L 3 , [ g L T 2 , [ A L 2 There are 5 dimensional quantities and 3 independent dimensions ( M,L,T ) , so that the number of dimensionless groupings is 2. Using the indicial method, the appropriate dimensional equation is [ h ]=[ ρ ] a 1 [ g ] a 2 [ A ] a 3 [ σ ] a 4 Substituting the dimensions for each quantity yields L = M a 1 L 3 a 1 L a 2 T 2 a 2 L 2 a 3 M a 4 T 2 a 4 = M a 1 + a 4 L 3 a 1 + a 2 +2 a 3 T 2 a 2 2 a 4 Thus, equating exponents, we arrive at the following three equations: 0= a 1 + a 4 1= 3 a 1 + a 2 +2 a 3 2 a 2 2 a 4

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154 CHAPTER 2. DIMENSIONAL ANALYSIS We can solve immediately for a 1 and a 2 from the first and third equations, viz., a 1 = a 4 and a 2 = a 4 Substituting into the second equation yields 1=3 a 4 a 4 +2 a 3 =2 a 3 a 4 = a 3 = 1 2 a 4 Substituting back into the dimensional equation, we have
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This note was uploaded on 02/09/2012 for the course AME 309 taught by Professor Phares during the Spring '06 term at USC.

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p0238 - 2.38. CHAPTER 2, PROBLEM 38 153 2.38 Chapter 2,...

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