Center of mass of a slender rod with variable density

  • Thread starter jonjacson
  • Start date
  • #1
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Hi to everybody

Homework Statement



I´ll show the problem with a picture:

2qtac8h.jpg


Homework Equations



L[itex]\overline{x}[/itex]=∫xc ρ dl

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?
 

Answers and Replies

  • #2
Doc Al
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Homework Equations



L[itex]\overline{x}[/itex]=∫xc ρ dl
Instead of L*xcm, the left hand side should be M*xcm. Where M is the mass of the rod.

I assumed too that the constant ρ0 at the equation for the density would not change anything, but maybe that is a mistake.
You can't just drop a constant, like you did when you integrated. :wink:
 
Last edited:
  • #3
426
35
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:

-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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