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 determine the energy/yield at a particular time after the explosion of a different bomb if you know the air density, time, and radius at that particular time .
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This note was uploaded on 02/05/2010 for the course CHEM 150A taught by Professor Muller during the Spring '10 term at University of California, Berkeley.

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

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