CENTER OF MASS of a circular conical surface

[Meister RaRo]

Homework Statement

Find the center of mass of a circular conical surface (empty cone) of height H.

Homework Equations

z(CM) = 1/M * int(z*dm)
x(CM)=y(CM)=0 (we've taken the origin of coordinates at the center of the base)

The Attempt at a Solution

This problem far exceeds my mathematical knowledge so I'll probably be talking nonsense but... here it goes:
let L be the slant height of the cone, R the radius of the circular base
dm= (M/pi*R*L)*dS
dS= 2pi*r*dz (where r is the "radius of each dS", r= R(H-z)/H)
We can now solve the integral, obtaining:
z(CM)= H^2/3L
But, given the problem wording, z(CM) should depend only upon H...!

Thank you for your help (and excuse my lousy English)!

 

[ozymandias]

Unless I've misunderstood your question, the result can definitely not depend on H alone.
Consider two cones of height H, one with a base of radius R1 and another with a base of radius R2.
Now take R1>>R2. It seems clear to me that cone #1 will have its center of mass well below that of cone #2.--------
Assaf
http://www.physicallyincorrect.com/"