Center of mass of a cone

  1. I need to find the center of mass of a cone with point facing downwards, of height H and radius R.

    Since the density is constant throughout and because of axial symmetry the center must be somewhere on the z-axis.

    I know from convention that this is H/4 but i need to derive this.


    Rcm = (intregral from 0 to H) of the change in radius

    this is where im stumped
    i did really bad in calculus

    could anyone help me?
     
  2. jcsd
  3. Doc Al

    Staff: Mentor

    Hint: Consider the cone as a stack of disks.
     
  4. Let Dv Be An Element In The Form Of A Disk That Cuts Through The Cone.

    The Radius Of The Disk Is (r / H) X.

    The Volume Equals The Area Of The Disk Times The Thickness.

    Dv = Pi[(r / H ) X] ^2
    Now Intergate From 0 To H

    X' = Int (x Dv) / Int Dv = 3/4 H
     
  5. okay so the biggest such disk would have volume pi*R^2*h

    what is the volume of the disk under that?
     
  6. lightgrav

    lightgrav 1,243
    Homework Helper

    the biggest *THIN* disk, at x = H, has radius r = xR/H,
    so its Volume = dV = pi R^2 dx.

    You need to integrate x from 0 to H .
     
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