# Thread: Cones - Center of Gravity, Center of Mass, etc.

1. Order of Kilopi
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## Mistake in "Mistake"?

Originally Posted by xylophobe
When I started writing the program I just did it for the
cylindrical (rectangular) portion (of segment 2 but the
mass of the triangle part of the frustum equals 4/3 of
that of the cylindrical part so I was missing over half
of the mass on my first go around.
Maybe I don't know what you're talking about, but
surely the mass of the triangular part of frustum 2
is much less than that of the cylindrical part. Isn't
it 1/3 the mass, not 4/3 the mass?

-- Jeff, in Minneapolis

2. No, that's what he's talking about, but it is 4/3.

One way to look at it: compare the cone of height 2h with the one of height h that forms its top. Since they have proportional dimensions, the smaller cone is 1/2^3 the volume of the whole cone. That means the frustrum is 7/8 of the entire volume, but the cylinder is only 3 times the size of the upper cone, or 3/8 of the entire volume. The rest ("triangular" part) is 4/8 of the total.

3. Order of Kilopi
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Originally Posted by grapes
No, that's what he's talking about, but it is 4/3.

One way to look at it: compare the cone of height 2h
with the one of height h that forms its top. Since they
have proportional dimensions, the smaller cone is 1/2^3
the volume of the whole cone. That means the frustrum
is 7/8 of the entire volume,
What?????

No no no no no. Is the Universe upside-down? Surely
the small cone is 1/4 the volume of the whole cone and
the frustum is 3/4 the volume of the whole cone.

-- Jeff, in Minneapolis

4. Originally Posted by Jeff Root
What?????

No no no no no. Is the Universe upside-down? Surely
the small cone is 1/4 the volume of the whole cone and
the frustum is 3/4 the volume of the whole cone.

-- Jeff, in Minneapolis
The universe isn't upside down, but it is three-dimensional. Are you calculating the volume of a cone or the area of a triangle?

5. Order of Kilopi
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Grapes,

The illustration below is what we're talking about,
isn't it? Given that this is the second segment, with
half the height of the full cone, the volume of the
outer part, which xylophobe is calling the triangular
part, can't be greater than the volume of the inner
cylindrical part.

-- Jeff, in Minneapolis

.

6. Originally Posted by Jeff Root
Grapes,

The illustration below is what we're talking about,
isn't it? Given that this is the second segment, with
half the height of the full cone, the volume of the
outer part, which xylophobe is calling the triangular
part, can't be greater than the volume of the inner
cylindrical part.

-- Jeff, in Minneapolis

.
Oh yes it can, and it is greater. I just did the number crunching myself and found that the volume of the outer part indeed is 4/3 times the cylinder. Perhaps counterintuitive, but my numbers don't lie.

7. Originally Posted by Jeff Root
Grapes,

The illustration below is what we're talking about,
isn't it? Given that this is the second segment, with
half the height of the full cone, the volume of the
outer part, which xylophobe is calling the triangular
part, can't be greater than the volume of the inner
cylindrical part.

-- Jeff, in Minneapolis

.
My math shows Grapes is right, Jeff. Using r as the radius at the top of your diagram, and h as the height of your diagram, I come up with 7/3*pi*r^2*h for the volume. The volume of the internal cylinder, of course, is pi*r^2*h, which is less than half of the total.

This also matches up with Grapes' numbers of 3/8 and 4/8 of the original entire cone (with the lopped-off top cone being the remaining 1/8).

8. Order of Kilopi
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Sean,

I calculated it twice, but it appears that both times I
thought I was done when I hadn't yet multiplied by 2
for the height of the full cone.

Grumblegrumblegrumble.

I seem to make that mistake a lot: Think I'm done
before I really am. Like this post.

-- Jeff, in Minneapolis

9. Order of Kilopi
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Doubling the length of a cone multiplies the volume by eight.

Yow!

-- Jeff, in Minneapolis

10. Originally Posted by grapes
One way to look at it: compare the cone of height 2h with the one of height h that forms its top. Since they have proportional dimensions, the smaller cone is 1/2^3 the volume of the whole cone.
Or, the larger cone is 2^3 times the size of the smaller cone.

11. Originally Posted by Jeff Root
Doubling the length of a cone multiplies the volume by eight.

Yow!

-- Jeff, in Minneapolis
No, doubling the length of a cone multiplies the volume by two. Doubling the radius of a cone multiplies the volume by four. Doubling both multiplies the volume by eight.

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