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Dimensional analysis - atomic bomb explosion radius

  1. Oct 11, 2014 #1
    1. The problem statement, all variables and given/known data:

    An atomic explosion can be approximated as the release of a large amount of energy ##E## from a point source. The explosion results in an expanding spherical fireball bounded by powerful shock wave. Let ##R## be the radius of the shock wave and assume that ##R=f(E,T,\rho_0, p_0)## where ##t## is the elapsed time after the explosion takes place, ##\rho_0## is the ambient air density and ##p_0## is the ambient air pressure. Using dimensional analysis show that ##R=\frac{Et^2}{\rho_0}g(\pi_1)## where ##\pi_1## is a dimensionless variable. Choose it so that it involves ##E## to a negative exponent. What is ##\pi_1##?

    2. Relevant equations

    ##R=f(E,T,\rho_0, p_0)##
    ##R=\frac{Et^2}{\rho_0}g(\pi_1)## where ##\pi_1## is a dimensionless variable

    3. The attempt at a solution

    The problem is that the relationship does not appear to be homogeneous. If ##\pi_1## is dimensionless, then the units of ##\frac{Et^2}{\rho_0}## must be the same as the units of ##R##. However,

    ##[E] = ML^2 T^{-2}##
    ##[t] = T##
    ##[\rho_0] = ML^{-3}##
    ##[p_0] = ML^{-1}T^{-2}##
    ##[R]=L##, where M is express in kilograms, T is in seconds, L is in meters.

    What is it that I'm not getting right here?

    On the other hand, I expressed ##\pi_1## as follows: ##\pi_1 = \frac{p_0 R^3}{E}##. Is this correct?

    Thank you!
     
  2. jcsd
  3. Oct 12, 2014 #2

    yes, it's correct. The dimension ##p_0 R^3## is same for E.
    ##ML^{-1}T^{-2}## * ##L^{3}## = ##ML^2T^{-2}##
     
    Last edited: Oct 12, 2014
  4. Oct 12, 2014 #3
    Thanks. But do you think the expression given in the problem statement is correct at all? The problem is that ##R## has units of ##L##, but ##\frac{Et^2}{\rho_0}## has units of ##L^{-5}##.
     
  5. Oct 12, 2014 #4
    unit of ##Et^2## is ##ML^2T^{-2} * T^2## then ##ML^2##
    so unit of ##R## is ##\frac{ML^2}{ML{-3}}## = ##L^5##
     
  6. Oct 12, 2014 #5
    Sorry, I meant ##L^5##. So then the expression is not quite homogeneous. Namely, the units of ##R## are ##L##, but the units of ##\frac{ET^2}{\rho_0}## are ##L^5##. Am I correct?
     
  7. Oct 12, 2014 #6
    yes , you are. Below file can be useful
    http://dspace.mit.edu/bitstream/handle/1721.1/42045/228875559.pdf?sequence=1 [Broken]
     
    Last edited by a moderator: May 7, 2017
  8. Oct 12, 2014 #7
    Thanks. So, the ##\frac{ET^2}{\rho_0}## should be raised to the power of 1/5, shouldn't it?
     
  9. Oct 12, 2014 #8
    yes, it's right
     
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