Homework Help: CENTER OF MASS of a circular conical surface

1. Dec 1, 2007

Meister RaRo

1. The problem statement, all variables and given/known data
Find the center of mass of a circular conical surface (empty cone) of height H.

2. Relevant 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)

3. 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)!!!

2. Dec 1, 2007

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.

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Assaf
http://www.physicallyincorrect.com/" [Broken]

Last edited by a moderator: May 3, 2017