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Midterm 1 Solution

Midterm 1 Solution - #1 From declassified pictures of an...

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#1. From declassified pictures of an atomic blast, Geoffrey Taylor was able to predict the energy, E, of the atomic bomb solely from measurements of the radii of the blast, r, at a particular time, t, after the blast and knowing the ambient air density, ρ , and pressure, P. a) Use the Buckingham Pi theorem with E, ρ , and t as core variables to determine relevant dimensionless parameters and write the relation between them in functional form. b) Because the energy is very large, for small times after the explosion the dimensionless parameter containing P is nearly zero and the functional dependence on this dimensionless parameter can be considered constant. Knowing this and using your result from part a), you want to determine the energy for a different bomb. What parameters would you either have to know or measure if you wanted to predict the energy of this bomb? Solution a) r t P ρ E M 0 0 1 1 1 L 1 0 -1 -3 2 T 0 1 -2 0 -2 5-3 = 2 dimensionless groups " 1 = rt a # b E c = L T [ ] a M L 3 $ % & ( ) b ML 2 T 2 $ % & ( ) c * a = + 2/5, b = 1/5, c = + 1/5 * " 1 = rt + 2/ 5 # 1/ 5 E + 1/ 5 " 2 = Pt d # e E f = M LT 2 T [ ] d M L 3 $ % & ( ) e ML 2 T 2 $ % & ( ) f * d = 6/5, e = + 3/5, f = + 2/5 * " 2 = Pt 6/ 5 # + 3/ 5 E + 2/ 5 " 1 = rt + 2/ 5 # 1/ 5 E + 1/ 5 = f ( " 2 ) = f ( Pt 6/ 5 # + 3/ 5 E + 2/ 5 ) b) The functional form becomes: " 1 = rt # 2/ 5 $ 1/ 5 E # 1/ 5 = f ( " 2 ) = cons tan t , so you can
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