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

  1. #121
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    Area doe not equal Volume

    Quote Originally Posted by ShinAce View Post
    ... Area is mass in this case so mass goes up by d^2 while gravity goes down by d^2. Unity.
    Quote Originally Posted by grapes View Post
    ...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.

  2. #122
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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. #123
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    Quote Originally Posted by xylophobe View Post
    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:
    Quote Originally Posted by xylophobe View Post
    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.

  4. #124
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    Okay

    Quote Originally Posted by grapes View Post
    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.

  5. #125
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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. #126
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    Quote Originally Posted by xylophobe View Post
    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.

  7. #127
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    Quote Originally Posted by ShinAce View Post
    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.


    Quote Originally Posted by ShinAce View Post
    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
    http://www.FreeMars.org/jeff/

    "I find astronomy very interesting, but I wouldn't if I thought we
    were just going to sit here and look." -- "Van Rijn"

    "The other planets? Well, they just happen to be there, but the
    point of rockets is to explore them!" -- Kai Yeves

  8. #128
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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.

    Click image for larger version. 

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  9. #129
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    Quote Originally Posted by xylophobe View Post
    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.

    Click image for larger version. 

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    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. #130
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    Quote Originally Posted by xylophobe View Post
    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.

  11. #131
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    Quote Originally Posted by grapes View Post
    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.

  12. #132
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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.

  13. #133
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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
    http://www.FreeMars.org/jeff/

    "I find astronomy very interesting, but I wouldn't if I thought we
    were just going to sit here and look." -- "Van Rijn"

    "The other planets? Well, they just happen to be there, but the
    point of rockets is to explore them!" -- Kai Yeves

  14. #134
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    Pappus's centroid theorem

    Quote Originally Posted by Jeff Root View Post
    ... 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. #135
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    Quote Originally Posted by xylophobe View Post
    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.

  16. #136
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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.

    Click image for larger version. 

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  17. #137
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    Composite Centroid

    Quote Originally Posted by Hornblower View Post
    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. #138
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    Quote Originally Posted by xylophobe View Post
    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.
    What is your answer about the gravitational effect?

  19. #139
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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
    http://www.FreeMars.org/jeff/

    "I find astronomy very interesting, but I wouldn't if I thought we
    were just going to sit here and look." -- "Van Rijn"

    "The other planets? Well, they just happen to be there, but the
    point of rockets is to explore them!" -- Kai Yeves

  20. #140
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    Mass Scales Proportionally to Distance

    Quote Originally Posted by grapes View Post
    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. #141
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    Quote Originally Posted by Jeff Root View Post
    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.
    Right about what?

  22. #142
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    Quote Originally Posted by xylophobe View Post
    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.

  23. #143
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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...

    Quote Originally Posted by ShinAce View Post
    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
    http://www.FreeMars.org/jeff/

    "I find astronomy very interesting, but I wouldn't if I thought we
    were just going to sit here and look." -- "Van Rijn"

    "The other planets? Well, they just happen to be there, but the
    point of rockets is to explore them!" -- Kai Yeves

  24. #144
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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.

  25. #145
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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).


    Click image for larger version. 

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  26. #146
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    Quote Originally Posted by xylophobe View Post
    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!
    Quote Originally Posted by grapes View Post
    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. #147
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    ok? {maybe not {{ the {{{ i mean my QUestion is ? can U (Um}? map the above
    integral form to a (Um2}? " Differential form "

    http://www.principiadiscordia.com/fo...g,1141630.html
    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. #148
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    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.
    Time wasted having fun is not time wasted - Lennon
    (John, not the other one.)

  29. #149
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    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.

  30. #150
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    Quote Originally Posted by Jeff Root View Post
    Quote Originally Posted by ShinAce View Post
    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.
    Quote Originally Posted by ShinAce View Post
    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

    http://www.bautforum.com/showthread....51#post2007151

    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.

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

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

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

    Quote Originally Posted by ShinAce View Post
    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.

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

    Quote Originally Posted by ShinAce View Post
    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.

    Quote Originally Posted by ShinAce View Post
    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.

    Quote Originally Posted by ShinAce View Post
    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.

    Quote Originally Posted by ShinAce View Post
    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???

    Quote Originally Posted by ShinAce View Post
    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???

    Quote Originally Posted by ShinAce View Post
    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
    sure about that?

    Quote Originally Posted by ShinAce View Post
    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
    http://www.FreeMars.org/jeff/

    "I find astronomy very interesting, but I wouldn't if I thought we
    were just going to sit here and look." -- "Van Rijn"

    "The other planets? Well, they just happen to be there, but the
    point of rockets is to explore them!" -- Kai Yeves

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