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?
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
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 .