Center of mass of a cone

  • Thread starter dowjonez
  • Start date
  • #1
22
0
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?
 

Answers and Replies

  • #2
Doc Al
Mentor
45,093
1,397
Hint: Consider the cone as a stack of disks.
 
  • #3
208
0
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
 
  • #4
22
0
okay so the biggest such disk would have volume pi*R^2*h

what is the volume of the disk under that?
 
  • #5
lightgrav
Homework Helper
1,248
30
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 .
 

Related Threads on Center of mass of a cone

  • Last Post
Replies
9
Views
10K
  • Last Post
Replies
4
Views
2K
  • Last Post
Replies
2
Views
4K
  • Last Post
Replies
3
Views
34K
Replies
15
Views
3K
Replies
9
Views
3K
Replies
3
Views
328
Replies
16
Views
253
  • Last Post
Replies
3
Views
695
  • Last Post
Replies
3
Views
4K
Top