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Homework Help: CENTER OF MASS of a circular conical surface

  1. Dec 1, 2007 #1
    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. jcsd
  3. Dec 1, 2007 #2
    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.

    http://www.physicallyincorrect.com/" [Broken]
    Last edited by a moderator: May 3, 2017
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