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

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## Area doe not equal Volume

Originally Posted by ShinAce
... Area is mass in this case so mass goes up by d^2 while gravity goes down by d^2. Unity.
Originally Posted by grapes
...If, as you say, the radius varies directly proportional with distance, then since mass is proportional to the square of the radius, mass must be proportional to the square of distance. No? Why would mass be directly proportional to diameter?
I hate this "area" argument!! The simple fact is that area does not equal mass nor does area equal volume nor does area equal gravity.

Check the units: when area (m2) is multiplied by density (kg/m3) we end up with kg/m which is non-sense.

I tried to show to ShinAce that the pizza argument is only applicable to cylinders with a fixed length but diameter varying with distance and now grapes is confused - we are talking about frustums not cylinders.

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Now this is comical. Grapes and i do these types of calculations often enough. I'm majoring in physics and 80% of what i do is setup integrals.

We have density, depth, and radius. We assume that each slice of pizza weighs the same. Then you take pi*radius^2 to get the area of the pizza. Multiply that area by depth and you have volume. Multiply that by density and you get mass. Density is mass per volume.

What we're doing is ignoring depth because we only consider pizzas with equal depth. Let's say they have a depth of one. Then all pizza areas have to be multiplied by one to get volume. Tada! That doesnt change the answer when u multiply by one.

This lets us compare areas as if they were mass equivalent. As long as density is constant everywhere, no fancy math is needed. Area is proportional to mass in this case.

3. Originally Posted by xylophobe
I hate this "area" argument!! The simple fact is that area does not equal mass nor does area equal volume nor does area equal gravity.

Check the units: when area (m2) is multiplied by density (kg/m3) we end up with kg/m which is non-sense.
I think we all agree with that.

When they said "area is mass", in context, they clearly meant that area is proportional to mass, by a constant in the case of a fixed depth of the "pizza".
I tried to show to ShinAce that the pizza argument is only applicable to cylinders with a fixed length but diameter varying with distance and now grapes is confused - we are talking about frustums not cylinders.
The one thing that I am confused about is what I mentioned in the last post:
Originally Posted by xylophobe
If I understand your description correctly you are describing cylinders with equal thicknesses (or "depths" as grapes put it) but the radius varies directly proportional with distance which then means that the volume/mass varies in a manner directly proportional to the diameter of the pizza-cylinders.
I'm pretty sure you're talking about cylinders here.

Why would you say that the mass of a cylinder be directly proportional to diameter? It's directly proportional to the square of the diameter.

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## Okay

Originally Posted by grapes
Why would you say that the mass of a cylinder be directly proportional to diameter? It's directly proportional to the square of the diameter.
Yes, you are correct, that was a mistake on my part.

As for the area equals mass argument in the context of the previous discussions about cylinders with fixed lengths then the comparison is valid but if we compare the cross-sections it is quite obvious that a frustum does not equal a cylinder.

The problem with the area-argument is that it confuses people and prevents them from actually doing the volume calculations.

Again, if you do the math per my first post using the familiar volume of a cone formula

V = pi()*r2*h/3

V3 = V1 - V2

and then compare with the distance you can see that the volume/mass of a frustum is not directly related to the distance-squared nor directly related to the distance.

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Isn't a frustrum just a slice of a cylinder?

If we're integrating infinitesimal slices, then a frustrum only changes the limits of integration. The cylinder is the full thing and a frustrum is a piece. I can take the hard work grapes put in, change the limits, and solve any frustrum i want.

His point stands regarding gravitational influence. He has every right to reduce it as he saw it.

6. Originally Posted by xylophobe
As for the area equals mass argument in the context of the previous discussions about cylinders with fixed lengths then the comparison is valid but if we compare the cross-sections it is quite obvious that a frustum does not equal a cylinder.
Did anyone really say otherwise?
The problem with the area-argument is that it confuses people and prevents them from actually doing the volume calculations.
I prefer to think that we confuse ourselves, and I don't really see how it *prevents* people from doing a calculation.
Again, if you do the math per my first post using the familiar volume of a cone formula

V = pi()*r2*h/3

V3 = V1 - V2

and then compare with the distance you can see that the volume/mass of a frustum is not directly related to the distance-squared nor directly related to the distance.

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Originally Posted by ShinAce
We have density, depth, and radius. We assume
that each slice of pizza weighs the same.
Only for a rather tricky definition of "weigh".

You assume that each slice of pizza has the same
density and the same depth. That gives larger pizza
slices greater mass, or "weight". But if by "weighs"
you mean force (per unit mass) applied at point P,
then yes, each slice weighs the same at point P.

Originally Posted by ShinAce
Isn't a frustrum just a slice of a cylinder?
A frustum is not a slice of a cylinder, it is a slice of a cone.
But my immediate intuitive impression is that frustums and
cylinders should work equally well. So should a variety of
other shapes in which the "depth" or thickness is the same
for the different slices.

-- Jeff, in Minneapolis

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## Mathematical Proof that Frustum Volumes are not Proportional to Distance

The image below is mathematical proof that the volume/mass of frustums are not proportional to distance.

What this means is that the mass per unit area is not equal for the segments (see post #100) which means that the gravitational influence upon point "P" differs for each segment of a cone.

The reason the volume/mass of frustums are not proportional to distance is because of the centroid of the triangular portion of each segment --- it is a constant r/3 and by definition constants are not variables so adding a constant to a variable eliminates the proportionality.

The math below is the application of Pappus's centroid theorem for determining volumes.

9. Originally Posted by xylophobe
The image below is mathematical proof that the volume/mass of frustums are not proportional to distance.

What this means is that the mass per unit area is not equal for the segments (see post #100) which means that the gravitational influence upon point "P" differs for each segment of a cone.

The reason the volume/mass of frustums are not proportional to distance is because of the centroid of the triangular portion of each segment --- it is a constant r/3 and by definition constants are not variables so adding a constant to a variable eliminates the proportionality.

The math below is the application of Pappus's centroid theorem for determining volumes.

You are dealing with a few thick frustums. If we take a large number of thin slices, as an exercise in rough and dirty calculus, the ratio of the incremental gravitational contributions at the vertex should approach a constant as a limit as h approaches zero.

10. Originally Posted by xylophobe
What this means is that the mass per unit area is not equal for the segments (see post #100) which means that the gravitational influence upon point "P" differs for each segment of a cone.
But the integral in post #101 seems to say that is wrong, that the gravitational effect at the apex is the same for each frustum having the same height.

It has nothing to do with mass bring proportional to any power of distance, it's just a calculation of the effect itself.

ETA: In other words, if you were put your segment one on the left, and any *one* of the other three on the right, at their appropriate distance, the gravitational acceleration at the apex due to the two segments would sum to zero.

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Originally Posted by grapes
The gravitational effect at the top point of the cone cannot be calculated by G times the mass of the cone, divided by the square of the distance to the COM/COG. It should be

if I've set it up right. Then you can divide by the mass of the cone times G

and take the inverse square root of that to get the effective distance
If I am reading your integrals correctly you are calculating for the whole cone because your limits start from zero for both integrals - to calculate the first frustum the limits should be between h and 2h and r and 2r

I am not an expert at integrals but I can solve for the volume of a frustum in only one line using an integral with limits between

0 and h for segment 1
h and 2h for segment 2
2h and 3h for segment 3
3h and 4h for segment 4

My results using integrals are the same as I receive using Pappus's centroid theorem - using two methods and getting the same answer is more persuasive than using one method alone.

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Thanks Jeff. I did mean cone instead of cylinder. Guess i overstudied for my calc exam. I just kept thinking how easy it is to solve this stuff in cylindrical coordinates.

+1 to concensus. When you have half of the cone on one side and the other half opposite, their gravity cancels. The division of half and half is measured by the height of the slices, not their mass. You can end up with more mass on one side, but most of that mass is further away.

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Hornblower,

Dividing the cone into negligibly-thin frustums may work to find
the solution, but I doubt it answer's xylophobe's objection. I think
some step of his analysis is just plain wrong. Which step? Is it
the assumption that the centroids of the triangular sections are
relevant?

-- Jeff, in Minneapolis

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## Pappus's centroid theorem

Originally Posted by Jeff Root
... I think
some step of his analysis is just plain wrong. Which step? Is it
the assumption that the centroids of the triangular sections are
relevant?

-- Jeff, in Minneapolis
You have to find the centroid of the component areas in order to use Pappus's centroid theorem - I used this method because it shows why the volume/mass of frustums are not proportional to the distance.

15. Originally Posted by xylophobe
You have to find the centroid of the component areas in order to use Pappus's centroid theorem - I used this method because it shows why the volume/mass of frustums are not proportional to the distance.
Has anyone said otherwise? If so, please direct me to it.

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## Correction

By the way: I did make a mistake with my page of calculations because I forgot to carry pi() to the answers so here is the fixed version.

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## Composite Centroid

Originally Posted by Hornblower
Has anyone said otherwise? If so, please direct me to it.
This link will get you to how to find a composite centroid for an area.

18. Originally Posted by xylophobe
If I am reading your integrals correctly you are calculating for the whole cone because your limits start from zero for both integrals - to calculate the first frustum the limits should be between h and 2h and r and 2r
The x limits would go from h to 2h, but the y limits would stay the same--zero up to a max of 2r
I am not an expert at integrals but I can solve for the volume of a frustum in only one line using an integral with limits between
I wasn't solving for mass or volume, I was calculating gravitational effect
My results using integrals are the same as I receive using Pappus's centroid theorem - using two methods and getting the same answer is more persuasive than using one method alone.

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Hornblower,

It seems apparent that I suggested otherwise, but I was writing
out of ignorance. If you agree that the centroids of the triangular
sections can be used to determine the total gravitational effect
of the slices, then I accept that you are both right. Though I'm
not going to study the Wikipedia article deeply enough to grok it.
Even though it is admirably brief and apparently concise.

Well... in order to write the above, I had to look at that section
of the article again. Although the math looks complex at first
glance, it is really very simple. Obviously it works for plane
figures. Maybe it works for solids, too.

-- Jeff, in Minneapolis

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## Mass Scales Proportionally to Distance

Originally Posted by grapes
I wasn't solving for mass or volume, I was calculating gravitational effect What is your answer about the gravitational effect?
I have an idea how to calculate the gravitational effect of the frustums but I do not have the integral skill to do so, yet.

My understanding is that the mass is supposed to scale up with increasing distance so that the increase in mass exactly balances the increase in distance - I do not see that with frustums but it is quite obvious for cones (segment 1 is a cone).

21. Originally Posted by Jeff Root
If you agree that the centroids of the triangular
sections can be used to determine the total gravitational effect
of the slices,
They cannot. Objects with the same mass and centroid can have different gravitational effects.
then I accept that you are both right.

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Originally Posted by xylophobe
I have an idea how to calculate the gravitational effect of the frustums but I do not have the integral skill to do so, yet.

My understanding is that the mass is supposed to scale up with increasing distance so that the increase in mass exactly balances the increase in distance - I do not see that with frustums but it is quite obvious for cones (segment 1 is a cone).
The integration is very helpful. If you want volume, replace everything inside the integral with the number 1. If you want mass, replace 1 with the density function. So far, we've stuck to a constant. In othet words, the material of the solid isn't wood in one case and metal in another. If you want gravitational effect, which grapes started with, you add the force expression to the integration. What you then find is what we've been arguing for a while now. That it doesn't matter if its a round cone, an elliptical cone or a slice of any type of pyramid. As long as the walls of that solid are projecting directly away from the origin, you get that any particular slice of equal height contributes an equal gravitational force.

There's another implication as we let the frustrum have infinite length, but it's too early to cover that.

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Grapes,

Should I reply to that? The smiley is telling me that I still
don't understand what's going on...

Originally Posted by ShinAce
As long as the walls of that solid are projecting directly away
from the origin, you get that any particular slice of equal height
contributes an equal gravitational force.
As I said twice above, my impression is that it should work
for many other shapes as well, in which the sides are not
projecting directly away from the origin. For example, your
cylindrical pizzas.

-- Jeff, in Minneapolis

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If we're talking about a cylinder in the sense of a pipe, then no. Each slice dx has equal mass and volume, but not gravitation influence. Only symmetrical slices cancel.

If we talk about a "cylindrical cone". Let's call it a line passing through and pivoting about the origin. Take this line and move it around. Once you get back to where you started, you'll have a shape that is both 'cylindrical' and 'conical'. The reason I call it cylindrical is because you would integrate it in cylindrical coordinates. These shapes obey the law that slices of equal thickness have equal gravitational influence. But then they have varying masses and volumes. As long as it's a shape you can create by pivoting that line through the origin, it doesn't matter if you later remove sections.

Such shapes include(the shapes are actually doubled, tip to tip): the cone with a circular base, the cone with an elliptical base, a pyramid of any base{triangle, square, hexagon, etc...). In plain english, the shapes are 'conical' but not necessarily with a circular base.

Straight old cylinders don't work. As you approach infinity, the distace is so large as to make the gravitational influence zero. If there was such a thing as infinity/2 , integrating from 0 to infinity/2 would not be equivalent to integrating from infinity/2 to infinity. For the cones, those integrals are equivalent.

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## Using the Familiar Formula

Here is an image that shows the volume/mass calculations that I did before only using the familiar cone volume formula = 1/3pi()r2h

It also includes the volume/mass divided by area which shows that the volumes/masses are not proportional to the distance.

ShinAce, In regard to your post #142 I would appreciate a more detailed description of the gravitational effect double-integral (by grapes) showing the parts and how it is calculated, I am trying to re-learn calculus.

I have been trying to give links and show my calculations so that others can follow and I appologize to Jeff Root for not explaining the importance of calculating the composite centroid for use with the Pappus calculation. The Pappus calculation is basically multiplying the total cross-sectional area by the distance that the composite centroid would travel and since it is a circle then it is simply the Y-value of the composite centroid, as the radius, used in the arc length formula = 2*pi()*r

The reason the mass of frustums do not scale proportionally with distance is that the "r" in the Pappus calculation has the constant r/3 added to it.

The reason frustums differ from cones is because cone cross-sections only have 3 lines and 3 defining parameters: h, r, r/3

h is the height of the cone
r is the radius of the base
r/3 is the y-location of the centroid for the area

whereas,

frustum cross-sections have 4 lines and 4 defining parameters: h, r1, r2, r-composite.

h is the height of the cone or frustum and for our discussion, to make is simple, is always equal for the cones and frustums (as shown in my pictures)
r1 is the front (small) face of the frustum
r2 is the back (large) face of the frustum
r-composite is the composite centroid y-location

I suspect that if the double-integral is calculated with the correct limits for x in regard to segment 2 (using the limits h and 2h) then the result will not equal the result for segment 3 (using the limits 2h and 3h) or segment 4 (using the limits 3h and 4h) or segment 1 (using the limits 0 and h).

26. Originally Posted by xylophobe
ShinAce, In regard to your post #142 I would appreciate a more detailed description of the gravitational effect double-integral (by grapes) showing the parts and how it is calculated, I am trying to re-learn calculus.
I'm happy to do that myself!
Originally Posted by grapes
The gravitational effect at the top point of the cone cannot be calculated by G times the mass of the cone, divided by the square of the distance to the COM/COG. It should be

G is the gravitational constant, delta is the density, 2pi y is the circumference of a ring which when multiplied by its cross-section dx dy gives the volume of an infinitesimal ring (multiplied by density gives mass), which is then multiplied by G and divided by distance squared to get the gravitational effect at the apex. However, opposite sides of the ring oppose each other, and cancel some of that force, leaving only the force parallel to the axis, which is found by multiplying by x divided by the distance.
I suspect that if the double-integral is calculated with the correct limits for x in regard to segment 2 (using the limits h and 2h) then the result will not equal the result for segment 3 (using the limits 2h and 3h) or segment 4 (using the limits 3h and 4h) or segment 1 (using the limits 0 and h).
The nice thing about the evaluation of that integral is that the y-integral is a constant! The x variable disappears:

Multiplying by the other four constants ( ) is still a constant, so the effect of the x-integral is just to multiply that resulting constant by the difference in the x limits. In other words, in your cases, h, each time. So, all four segments have the same answer.

27. ok? {maybe not {{ the {{{ i mean my QUestion is ? can U (Um}? map the above
integral form to a (Um2}? " Differential form "

riday 1/27 / 13 Chuen 2815 24/22.5 = .3sB
Integral form ...-> Differential form ...-> Matrix form
so the Question becomes ? wiLL i ever return?
i donno but i did find appendix I, so i may as
well try: starting from the integral form page 225
_/` E.dl = -dOb/dt & _/`B.dl=uo[i+eodoE/dt]
---------------------------------------------
E { electric field
B { magnetic field
O { flux
i { current

No, i have no idea
this was in January ? it is now April ?/?

28. My only contribution to the thread is to say that the reason Indian tepees are conical is because a cone is a shape that wouldn't blow over in the prairie winds.

29. Hopefully the etc. in the title makes the following valid comments.
*- Any cone (- the base) will unravel to a section of a circle
*- IIRC, the area of any such subsection is equal (radius x arc-length)/2
*- The conical aspect of eggs prevents them from rolling off shallow slopes. There is a correlation between the conical-ness of bird's eggs and how high they nest on cliffs. If a cone doesn't slip it will simply roll around the apex and reach a stable equilibrium which will stop any further rolling.

ETA:From Archimedes:
Any sphere is (in respect of solid content) four times the cone with base equal to a great circle of the sphere and height equal to its radius
Source
Last edited by a1call; 2012-Apr-15 at 05:59 AM.

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Originally Posted by Jeff Root
Originally Posted by ShinAce
As long as the walls of that solid are projecting directly away
from the origin, you get that any particular slice of equal height
contributes an equal gravitational force.
As I said twice above, my impression is that it should work
for many other shapes as well, in which the sides are not
projecting directly away from the origin. For example, your
cylindrical pizzas.
Originally Posted by ShinAce
If we're talking about a cylinder in the sense of a pipe,
then no. Each slice dx has equal mass and volume, but
not gravitation influence. Only symmetrical slices cancel.
I'm totally confused.

Everybody knows what a cylinder is. But "a cylinder in the
sense of a pipe" makes it sound as though a hole is a feature
of what we're talking about. While a hole appears to be a
feature of xylophobe's analysis of frustums, it certainly isn't a
feature I had in mind.

Your idealized pizzas, which you suggested in post #114

are cylindrical, aren't they? They have parallel sides, not sloped.
They don't have holes in them, do they?

They do not have "equal mass and volume". The farther the
pizza is from the origin, the larger its diameter, volume, and
mass.

Originally Posted by ShinAce
If we talk about a "cylindrical cone".
A what??? What the heck kind of terminology is that???

Originally Posted by ShinAce
Let's call it a line passing through and pivoting about the origin.
That sounds like the surface of a cone.

Originally Posted by ShinAce
Take this line and move it around.
A second time? Or are you re-stating the "pivoting" of the
preceeding sentence?

Originally Posted by ShinAce
Once you get back to where you started, you'll have a shape
that is both 'cylindrical' and 'conical'.
No. I haven't a clue. Gimme a diagram.

Originally Posted by ShinAce
The reason I call it cylindrical is because you would integrate
it in cylindrical coordinates.
I would? Maybe if I knew what that meant.

Originally Posted by ShinAce
These shapes obey the law that slices of equal thickness
have equal gravitational influence.
Well, you appear to be talking about the right subject, so I'm
trying to follow along.

But then they have varying masses and volumes.[/quote]
Yes, naturally.

Originally Posted by ShinAce
As long as it's a shape you can create by pivoting that line
through the origin, it doesn't matter if you later remove sections.
That seems to be a description of what xylophobe is doing in
his diagrams of two cones divided up into frustums, and makes
sense to me, so maybe I'm not completely lost.

Originally Posted by ShinAce
Such shapes include (the shapes are actually doubled, tip to tip):
the cone with a circular base, the cone with an elliptical base, a
pyramid of any base{triangle, square, hexagon, etc...). In plain
english, the shapes are 'conical' but not necessarily with a circular
base.
I think I understand that. I vaguely recall trying unsuccessfully
to find a general term for such a family of shapes, myself, a few
years ago.

Originally Posted by ShinAce
Straight old cylinders don't work.
I would think they would, although perhaps the distances would
be described slightly differently: Rather than measuring to the
near surface or the far surface or the center of mass, it might
be necessary to measure to the center of gravity. Yes???

Originally Posted by ShinAce
As you approach infinity, the distace is so large as to make the
gravitational influence zero.
What? Are we talking about the same thing?? The farther
away the disk is, the bigger it is. The gravitational influence
on the origin point should be the same from all disks, nomatter
how far away. Yes???

Originally Posted by ShinAce
If there was such a thing as infinity/2 , integrating from 0 to
infinity/2 would not be equivalent to integrating from infinity/2
to infinity.
Not that I understand what you are saying there, but are you

Originally Posted by ShinAce
For the cones, those integrals are equivalent.
Okay. How about you try to describe these integrals for me
in simple terms? (Although both you and grapes probably
already did so more than once, above...) Then apply it to
your "cylindrical cone". By which I'll now guess you simply
meant a stack of cylinders whose envelope is a cone.
Is that what you meant??????? That would make so much
sense. Please tell me that's what you meant!

And / or draw a diagram! Diagrams rule!

-- Jeff, in Minneapolis

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