Deriving the Center of Mass of a Cone with Point Facing Downwards

[dowjonez]

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 I am stumped
i did really bad in calculus

could anyone help me?

 

[Doc Al]

Hint: Consider the cone as a stack of disks.

 

[mathmike]

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

 

[dowjonez]

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

what is the volume of the disk under that?

 

[lightgrav]

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 .