How Do I Find the Centroid of a Cone Using Integration?

[jdub1989]

Homework Statement

Determine the centroid of the conical volume using integration. Height h and radius r. No numbers given.

Homework Equations

V = ∫dV = (from 0 to h)∫∏r²dz

The Attempt at a Solution

I know from looking around in my book where zbar is (xbar at zero and ybar at zero) but need to find it using integration.
I am stuck. I know that I need to find r in terms of z so that I can integrate it.
I also know that I need to use similar triangles to find the relation between z & r, but what similar triangles would I use? I think one would be the relation between r and h, but what's the relation involving dz?

 

[kurt.physics]

I think I am missing something obvious.EDIT: New info I foundAt the top of the cone, z=h and r=0At the base of the cone, z=0 and r=rSo, how do I use this information to find the equation of r in terms of z?A:You have the correct integral for the volume of a cone. To find the centroid, you need to use the equations for moment of inertia:$$x = \frac{1}{V}\int_0^h \int_0^{2\pi} \int_0^r r^2 r \cos \theta \ dz \ dr \ d\theta \ dz $$$$y = \frac{1}{V}\int_0^h \int_0^{2\pi} \int_0^r r^2 r \sin \theta \ dz \ dr \ d\theta \ dz $$$$z = \frac{1}{V}\int_0^h \int_0^{2\pi} \int_0^r r^2 \ dz \ dr \ d\theta \ dz $$And solve each one for x, y, and z.