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Problems with Dimensional Analysis.

  1. Aug 15, 2011 #1
    1. The problem statement, all variables and given/known data

    1) A string is stretched between two walls. When its plucked, a transverse wave travels from one end to the other. The length of the string is l, the mass of the string is m. There is a tensional force of F newtons present in the string. The velocity of the resultant wave is v m/s. It is reasonable to assume that there is some relationship between the wave velocity and the other parameters such as:
    v=k.F^x.l^y.m^z (where k is some dimensionless constant)

    Using dimensional analysis, determine the likely relationship.

    2) A sphere of radius a is dropped into a viscous liquid with a coefficient of viscosity n and its velocity at an instant is v. The frictional force can be partly found by dimensional analysis.

    F=k.a^x.n^y.v^z (where k is a dimensionless constant)

    The dimensions of the variables are:
    [F] = MLT^-2
    [a] = Lk
    [n] = ML^-1T^-1
    [v] = LT^-1

    Use dimensional analysis to find a likely relationship between the variables.


    3. The attempt at a solution
    This isn't coursework or homework, I have just been looking at past papers and a question similar to the ones I've posted always comes up. Most questions, I have some sort of an idea about, but dimensional analysis questions always stump me. Any help with methods on doing these types of questions would be appreciated.
     
  2. jcsd
  3. Aug 16, 2011 #2

    cepheid

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    Gold Member

    At this point it's just algebra. For instance, in your second problem, you know that the left-hand side must have dimensions of force, and so you just tweak the right-hand side until it matches:

    (I'm going to assume that [a] = L, since "Lk" doesn't make sense to me).

    MLT-2 = kLx(ML-1T-1)y(LT-1)z

    It helps if you first combine powers that have the same base:

    MLT-2 = kLx-y+zMyT-y-z

    Now you just have to match exponents up by inspection. In order for the right-hand side to have the same dimensions as the left, it must be true that:

    x-y+z = 1

    y = 1

    -y-z = -2

    substituting the second equation into the third, you get:

    -1 - z = -2

    z = 1

    substituting this into the first equation, you get:

    x - 1 +1 = 1

    x = 1

    Therefore, it seems that the answer is:

    F = kanv

    which might be plausible. I imagine that the solution method to most other problems of this type would be very similar.
     
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