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Center of mass of a cone

by dowjonez
Tags: cone, mass
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Oct18-05, 07:37 PM
P: 22
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?
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Doc Al
Oct18-05, 07:54 PM
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P: 41,477
Hint: Consider the cone as a stack of disks.
Oct18-05, 08:00 PM
P: 211
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

Oct18-05, 08:00 PM
P: 22
Center of mass of a cone

okay so the biggest such disk would have volume pi*R^2*h

what is the volume of the disk under that?
Oct18-05, 08:36 PM
HW Helper
lightgrav's Avatar
P: 1,117
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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