p0270 - 5 We can solve immediately for a 1 and a 2 , viz.,...

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2.70. CHAPTER 2, PROBLEM 70 207 2.70 Chapter 2, Problem 70 The dimensional quantities and their dimensions are [ h d ]= L, [ d ]= L, [ N ]= 1 T , [ Q ]= L 3 T , [ ν ]= L 2 T , [ g ]= L T 2 There are 6 dimensional quantities and 2 independent dimensions ( L, T ) ,sot h a tth e number of dimensionless groupings is 4. The appropriate dimensional equation is [ h d ]=[ d ] a 1 [ N ] a 2 [ Q ] a 3 [ ν ] a 4 [ g ] a 5 Substituting the dimensions for each quantity yields L = L a 1 T a 2 L 3 a 3 T a 3 L 2 a 4 T a 4 L a 5 T 2 a 5 = L a 1 +3 a 3 +2 a 4 + a 5 T a 2 a 3 a 4 2 a 5 Thus, equating exponents, we arrive at the following two equations: 1= a 1 +3 a 3 +2 a 4 + a 5 0= a 2 a 3 a
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Unformatted text preview: 5 We can solve immediately for a 1 and a 2 , viz., a 1 = 1 3 a 3 2 a 4 a 5 a 2 = a 3 a 4 2 a 5 Substituting back into the dimensional equation, we have [ h d ] = [ d ] 1 3 a 3 2 a 4 a 5 [ N ] a 3 a 4 2 a 5 [ Q ] a 3 [ ] a 4 [ g ] a 5 = [ d ] } Q Nd 3 ] a 3 } Nd 2 ] a 4 } g N 2 d ] a 5 Therefore, the dimensionless groupings are: h d d , Q Nd 3 , Nd 2 , g N 2 d...
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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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