1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

A question on Dimensional analysis

  1. Jun 12, 2014 #1
    This is actually an example from Physics for Scientists and Engineers by Serway. I am confused about the way they solved it.

    1. The problem statement, all variables and given/known data

    Suppose we are told that acceleration a of a particle moving with uniform speed v in a circle of radius r is proportional to some power of r, say r^m, and some power of v, say v^m. Determine the values of n and m and write the simplest form of an equation for the acceleration.

    2. Relevant equations

    The problem is solved in the book. But I dont understand the solution properly.This problem was solved in the example as below:

    [tex]a = k(r^n)(v^m)[/tex][tex]L/T^2 = (L^n)(L/T)^m = L^{n+m}/T^m[/tex][tex]n+m = 1 [/tex][tex] m = 2[/tex][tex]n = -1[/tex][tex]a = kr^{-1}v^2[/tex]

    3. The attempt at a solution

    From the question we don't know if k has any dimension or not. So how did they (in the second line) write [tex]L/T^2=(L^n)(L/T)^m ?[/tex] Where did the k go??
     
  2. jcsd
  3. Jun 12, 2014 #2

    Nathanael

    User Avatar
    Homework Helper

    k is by definition dimensionless. It's the constant of proportionality (meaning it's a constant that changes it from being "proportional" to being "equal")


    The definition of "a" being proportional to "b" is a=kb where k is some constant (hence it's a pure, dimensionless, number)
     
    Last edited: Jun 12, 2014
  4. Jun 12, 2014 #3
    OOPPss... Suddenly I understand. k is not anything like length mass or time, right? Maybe thats the reason why they left out k?
     
  5. Jun 12, 2014 #4
    Wait, I have another question. In law of gravitation we know F = GmM/r^2 Here G is a proportionality constant, isn't it? Then how come it is not a pure number (ie it has units like) 6.67×10^−11 N·(m/kg)2 ?
     
    Last edited by a moderator: Jun 12, 2014
  6. Jun 12, 2014 #5

    Nathanael

    User Avatar
    Homework Helper

    Correct it's a unitless number. (I think such a number is ofted called a "pure number")

    EDIT:
    Good question. I don't know a satisfactory answer. It seems I was wrong when I said "k is by definition dimensionless," I suppose it doesn't necessarily have to be dimensionless.

    (Which of course brings us back to your original question, how do we know k is dimensionless in your original post?)

    Sorry about the misinformation, I'm not sure why it must be dimensionless in your OP.
     
    Last edited: Jun 12, 2014
  7. Jun 12, 2014 #6

    Nathanael

    User Avatar
    Homework Helper

    Perhaps it's because they asked for "the simplest form of an equation for the acceleration."

    (Or perhaps it's just to avoid the infinite possibility that comes with k being of choose-able units)
     
  8. Jun 12, 2014 #7
    It never seems to be the case. Serway's book describes it as a general procedure. Here is the excerpt from Serway's book:(as this page can be previewed in amazon; I think it never breaks any copyright)

    Y3YP2Rd.png

    Now this certainly works for x ∞ (a^n)(t^m); but what happens when we apply this for, say - the law of gravitation? Here, F ∞ Mm/r^2 certainly doesn't pass the test of dimensional analysis (?). I am confused!
     
  9. Jun 12, 2014 #8

    SteamKing

    User Avatar
    Staff Emeritus
    Science Advisor
    Homework Helper

    Well, why don't you apply dimensional analysis to the law of gravitation? Hint: not every constant of proportionality is unitless, like the π in A = π r[itex]^{2}[/itex] for the area of a circle. And besides, 'constant' means the value doesn't change; it doesn't mean there are no units.
     
  10. Jun 12, 2014 #9
    Thanks.
    So I conclude - dimensional analysis works for equations; but it will work for proportionality only if the constant is unit less. (The book was a bit confusing.)
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted



Similar Discussions: A question on Dimensional analysis
Loading...