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Center of mass of a slender rod with variable density

  1. May 10, 2013 #1
    Hi to everybody

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

    I´ll show the problem with a picture:


    2. Relevant equations

    L[itex]\overline{x}[/itex]=∫xc ρ dl

    3. The attempt at a solution

    Well the total lenght of the rod is 1 feet, I only need to calculate the integral.

    The moment xc of a differential element of mass of the rod is the distance x to the y axis, and the density is known so:

    ∫x * (1-x/2) dx , from 0 to 1, the result for me is:

    1 * [itex]\overline{x}[/itex] = 1/3

    So [itex]\overline{x}[/itex]= 1/3;

    Unfortunately the result is 4/9, I can´t see where are my mistakes, maybe I´m not using the proper arm lenght or something.

    The problem doesn´t give you any coordinate system, only the x axis, I assume that the y axis is orthogonal, and the equation for the moments around that axis gives you the x center of mass.

    I suppose that nothing would change if I had used another coordinate system, Am I right?

    I assumed too that the constant ρ0 at the equation for the density would not change anything, but maybe that is a mistake.

    Where are my mistakes?
  2. jcsd
  3. May 10, 2013 #2

    Doc Al

    User Avatar

    Staff: Mentor

    Instead of L*xcm, the left hand side should be M*xcm. Where M is the mass of the rod.

    You can't just drop a constant, like you did when you integrated. :wink:
    Last edited: May 10, 2013
  4. May 10, 2013 #3
    -Ups, you are right I was used to use the equation of the centroids with constant density and I wrote the wrong equation.

    -I see, they cancel after integrating on both parts of the equation.

    Now I get the right result.

    Thank you very much!!!
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