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Dimensional consistency problem, need help

  1. Mar 27, 2014 #1
    Beginning student striving to learn. The problem was generated by people from the University of Winnipeg. Source: http://theory.uwinnipeg.ca/physics/ (this website was actually suggested by a user on the forums who suggested utilizing it as a learning source).


    1. The problem statement, all variables and given/known data
    Problem is, quote:

    The period of a simple pendulum, defined as the time for one complete oscillation, is measured in time units and is given by:

    T = 2∏√l/g

    where,l is the length of the pendulum and g is the acceleration due to gravity, in units of length divided by time squared. Show that this equation is dimensionally consistent; that is, show that the right hand side of this equation gives units of time.


    2. Relevant equations
    The solution is actually given. The full equation was as follows:

    [2∏√l/g]= √L/L/t2. This all equals: √t2 which = T


    3. The attempt at a solution

    Given the description which is italicized, I understood how they got: √L/L/t2 which ultimately equals T, but I do not understand how "2∏" disappeared so easily in the process of equating T.

    My attempt: The italicized description already tells us that "g"= L/t2. Given this information, dividing "L" by L/t2 will result in t2 (all under the square root symbol). Then, the √t2 is T. I get stuck on the conversion process when it comes to 2∏ however.

    I've thought about it for several minutes, and I still can't understand how to mathematically describe T and the aforementioned equation including 2∏. Can anyone help? Thank you for your time.
     
    Last edited: Mar 27, 2014
  2. jcsd
  3. Mar 27, 2014 #2

    NascentOxygen

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    Staff: Mentor

    Pi is a constant, it has no units. It is dimensionless. It is just a scaling factor, likewise the 2.
     
  4. Mar 27, 2014 #3

    This is hard for me to understand, and I probably need to enhance my mathematical skills as it relates to physics. However, during the entire time of trying to make sense of the equivalence, I kept thinking about multiplying 2 times 3.14 or something like that, and connect it to the rest of the equation. This is why I was stuck.

    I really appreciate your input, but if 2∏ was ultimately dimensionless to begin with, why was it put in the equation? Thanks for your time.
     
  5. Mar 27, 2014 #4

    SteamKing

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    Staff Emeritus
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    Homework Helper

    When you do dimensional analysis on equations in general, you must learn to distinguish any constants which might be contained in the equation from the parts of the equation which might contain units. Not all of the problems you encounter will be 'pre-digested' for your convenience.
     
  6. Mar 27, 2014 #5
    I think I understand what you're saying, but that doesn't really answer my question. Why were the constants put there in the first place if they were ultimately irrelevant in the conversion process?

    I mean, another conversion problem was posed on the website related to dimensional analysis (which dealt with converting miles into meters into kilometers), and I had no problem with that whatsoever.

    Yet, with this problem it was different. I don't wan't things necessarily pre-digested for me, but only an understanding of things...in the same way that someone would like an understanding as to why ∏=3.14592...; or something to that effect.
     
  7. Mar 27, 2014 #6

    NascentOxygen

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    Staff: Mentor

    Dimensionless constants are necessary in some formulae to make the formula correct, so that it gives the right answer. They are a scaling factor.

    Pi is the ratio of a circle's circumference to its diameter, so it is metres divided by metres, these cancel leaving no units. You have to recognize in formulae those constants which have no units, and dismiss them when checking dimensions.

    You may recognize the formula F=Gm1 m2 / r2

    here G is a constant, but it is not dimensionless, it has units.
     
    Last edited: Mar 27, 2014
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