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Homework Help: Dimensional analysis to determine unknown exponents

  1. Oct 25, 2008 #1
    1. The problem statement, all variables and given/known data
    1. Use the method of Dimensional Analysis to show that the unknown exponents in Equation (1) are l=-1/2, m=-1, and n=1/2.

    2. Relevant equations
    Equation (1) is [tex]\lambda = k \mu ^{l} f ^{m} T^{n}[/tex]

    [tex]\lambda[/tex] is the wavelength;
    f is the frequency of the sound;
    T is the tension in the string;
    [tex]\mu[/tex] is the mass per unit length of the string.
    k is a dimensionless constant.

    3. The attempt at a solution
    The dimensions for the above terms should be:
    [tex]\lambda = [L][/tex] (Simple enough)
    [tex]f=[L] ^{-1}[/tex] (Since the frequency is the inverse of time. Is this correct?)
    [tex]T=[M][L][T] ^{-2}[/tex] (Since the tension in the rope is just the force exerted on it, right?)
    [tex]\mu = [M][L] ^{-1}[/tex] (Since it is the mass per unit length)

    Which gives the dimensional equation as:
    [tex][L]=([M] \cdot [L]^{-1}) ^{l} \cdot ([T] ^{-1}) ^{m} \cdot ([M] \cdot [L] \cdot [T]^{-2})^{n}[/tex]

    Which can be used to make equations for [L], [T], and [M], respectively:

    1=-1l + 1n ([L]) (i)
    0=-1m - 2n ([T]) (ii)
    0=1l + 1n ([M]) (iii)

    And from here I don't know where to go. If I manipulate (ii) to state n in terms of m, I get n=-1/2m. But where do I go from here? I need to solve these three equations simultaneously?
    Last edited: Oct 25, 2008
  2. jcsd
  3. Oct 25, 2008 #2
    Oh, I got it. Sorry, I keep doing this with my posts here. XD
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