centroid of cylindrical cone

zandria

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

Determine the centroid of volume for a right circular cone with base diameter of 100mm and an altitude of 200mm.

2. Relevant equations

I know that if the my xy-plane is parallel to the base of the cylindrical cone then the x and y coordinates of the centroid must be zero and therefore I only need to find the z coordinate of the centroid.

The equation I am using is

[tex]z_c = \frac{1}{M} \int_{body} z dm[/tex]

where M is the total mass and [tex]dm = \rho dV[/tex]

3. The attempt at a solution

I am trying to use cylindrical coordinates but I think my limits of integration are incorrect. I have tried to solve the integral above with the following limits.

[tex]0<\theta<2\pi [/tex]

[tex]0<r<50 [/tex]

[tex]0<z<(200-r/4)[/tex]

I'm not sure if the limits for the z coordinate is correct. Am I on the right path?

LowlyPion

Well, I wouldn't worry with polar coordinates, because you are dealing with basically a stack of disks aren't you?

They are each have a weight of ρ*πr²

Exploit then the fact that r is a function of z, and your integral should be pretty straight forward shouldn't it?

zandria

Thank you. I was essentially doing the right thing on my first try before I changed everything, but I made an algebra mistake when trying to use cylindrical coordinates. Thanks for the short cut ... less room for stupid mistakes.