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String with variable density

  1. Sep 7, 2013 #1
    1. The problem statement, all variables and given/known data
    The linear mass density in a string is given by μ = μ0[1 + cos(x/R)] where R is a constant. If one averages this density over the large size L it becomes uniform: <μ> = μ0, where <…> means averaging. What is the minimum size L (in terms of R) such that the density can be considered uniform with an error less than 1% ?

    2. Relevant equations

    3. The attempt at a solution

    So I intergrate with respect to dx over the range o to L then divide by L because I'm averaging and what I get is U+UR/L*sin(L/R). However this is my problem. The initial mass density makes no sense. When x=R*pi the density is zero. How can the density be zero on a freaking string. That makes no sense. Besides that I don't now what is meant by error. Should I equal the U+UR/L*sin(L/R) to .99U then solve?
  2. jcsd
  3. Sep 7, 2013 #2
    I appears in my title I messed up. I mean variable density.

    [Moderator's note: thread title has been corrected by Redbelly98]
    Last edited by a moderator: Sep 8, 2013
  4. Sep 8, 2013 #3


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    You're right that it can't be zero on a real string. Better to think of it as negligibly small compared to the average, and calling it zero is an approximation.

    Yes. Ideally, it should be solved twice, using both 1.01U and 0.99U.
  5. Sep 8, 2013 #4
    Don't forget that the maximum and minimum values that sin(L/R) can take on are +1 and -1.

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