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Find values of p & q in v=K(lambda^p)(g^q)

  1. Aug 27, 2011 #1
    1. The problem statement, all variables and given/known data
    The speed of ocean waves depends on their wavelength lambda (measured in meters) and the gravitational field strength g (measured in m/s^2) in this way:
    v=K(lambda^p)(g^q)
    where K is a dimensionless constant. Find the values of the exponents p and q.


    2. Relevant equations



    3. The attempt at a solution
    P=(ln(v/k)-qlng)/(lnlambda)
    q=(ln(v/k)/(lambda^p))/(lng)
     
  2. jcsd
  3. Aug 27, 2011 #2

    PeterO

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    Homework Helper

    this is a units or dimension question.

    Suppose p was 3 and q was 5

    then V = K.L^3.g^5 [it was shorter to type L than spell out lambda]

    Look at the units.
    L^3 would have units m^3
    g^5 would have units m^5.s^-10

    combines that makes m^8.s^-10

    But this is velocity!!!! so the units would be m.s^-1 so 3&5 are certainly not the values.

    What should they be?
     
  4. Aug 28, 2011 #3
    Am I looking for actual numbers?
    The way I understand it, is that how quickly (how many meters per second) the crest of a wave moves across a distance depends on how far apart the crests are (in meters), then multiplied by the gravitational pull. The gravitational pull is measured in meters/second^2. I don't really understand the idea of gravitational pull being measured in meters per second^2. Also I don't understand why wavelength and gravitational pull are exponentiated.
    You say that this is a dimensions or units question, which means I need to convert from one unit to another I guess, but I don't see what it is I am converting. We just have meters, and meters over seconds, and the answer is supposed to be meters over seconds, so what is the problem?

    Thank you for your help thus far,
    -A
     
  5. Aug 28, 2011 #4

    PeterO

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    Homework Helper

    Acceleration units can be confusing, because of the way we just bunch similar units together - that is why the m/s^2. For understanding it is not metres per second squared, even though it is written like that.

    If you used a stop watch in a car, you might measure that it reached 60 km/h in 5 seconds.
    That represents an average acceleration of 12 (km/h)/s or 12 kilometres per hour per second

    I will use a comma to show where we pause when reading that:

    Kilometres per hour, per second.
    Seems clear but if we pause in the wrong place we would say

    kilometres, per hour per second which sound like gobbledy-gook.

    The real problem arises if the cars speedometer was calibrates in metres per second instead of kilometres per hour.
    You might measure that the car accelerated to 15 m/s in 5 seconds.

    That would mean an acceleration averaging 3 m/s each second - or 3 metres per second per second.
    When written in english, with the comma [pause] for emphasis that is

    3 metres per second, per second

    However in symbols we write that m/s^2 the exponent indicating per second occurred twice.

    If you multiply wavelength [m] by g [m/s^2] we get dimension/units of m^2/s^2, which can be written as (m/s)^2
    That means lambda x g gives the square of speed.
    to get just speed, we need the square root of that - and square root is shown as ^0.5

    eg 9^0.5 = 3 [or -3]

    so if the exponents in the original formula were both 0.5, the dimensions would be fine.
     
    Last edited: Aug 28, 2011
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