1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

How do I find the center of mass of a cone?

  1. Nov 7, 2011 #1
    Im trying to figure out the center of mass of a cone for this research I'm doing. How do I find the center of mass of a cone?
     
  2. jcsd
  3. Nov 7, 2011 #2
    any particular reason you can't research that too?
     
  4. Nov 7, 2011 #3

    phinds

    User Avatar
    Gold Member
    2016 Award

    take a cross-section though the axis. how would you find the center of mass of the resulting triangle?
     
  5. Nov 7, 2011 #4
    because I came across something called the Riemann Stieltjes integral and it is making no sense to me
     
  6. Nov 8, 2011 #5

    phinds

    User Avatar
    Gold Member
    2016 Award

    So did my question not make sense to you? Finding the center of gravity of a cone is exceedingly trivial and you do not need anything but the Pythagorean theorem. You certainly don't need any calculus, although I'm sure there are lots of hard ways to do it if you WANT to do it a hard way.
     
  7. Nov 8, 2011 #6

    HallsofIvy

    User Avatar
    Staff Emeritus
    Science Advisor

  8. Nov 8, 2011 #7

    phinds

    User Avatar
    Gold Member
    2016 Award

  9. Nov 8, 2011 #8

    HallsofIvy

    User Avatar
    Staff Emeritus
    Science Advisor

    No, I never said that. To determine the centroid of a general figure, you certainly need Calculus. But, you can then use a formula for a given type of figure- there is a standard formula for the centroid of a cone. Perhaps the OP could just look that up.
     
    Last edited: Nov 8, 2011
  10. Nov 8, 2011 #9

    phinds

    User Avatar
    Gold Member
    2016 Award

    agreed
     
  11. Nov 9, 2011 #10
    phinds,
    could you elaborate on your method of using only the Pythagorean theorem? I'm not getting it.
     
  12. Nov 9, 2011 #11

    phinds

    User Avatar
    Gold Member
    2016 Award

    The center of gravity of a cone is at the exact same spot as the center of the area of a triangle created if you pass a plane through the axis of the cone. Divide that triangle into an upper triangle (I'm thinking of the cone pointing downward) and a lower trapazoid, with the areas of the two being equal. Pythagorous (and an understanding of similar triangles) will give you the rest.
     
  13. Nov 9, 2011 #12
    I think you are right if you are considering a hollow cone. I was assuming the OP asked about a solid cone, but of course I might be wrong.

    For me, it is not very intuitive that the cross-sectional triangle trick will work if a hollow cone is considered. How did you come to that conclusion?
     
  14. Nov 9, 2011 #13

    phinds

    User Avatar
    Gold Member
    2016 Award

    I have no idea why you think the cone should be hollow for this to work; I'm assuming a solid cone. If the upper triangle has the same area as the lower trapezoid then will not the resulting rotated figures have the same volumes?

    OH ... oops, maybe they don't
     
  15. Nov 10, 2011 #14

    phinds

    User Avatar
    Gold Member
    2016 Award

    LATER: I checked it out and it does work out correctly in this case.
     
  16. Nov 10, 2011 #15
    The median of an isosceles triangle is H/3 while the CoM of a cone is at H/4. I'm not quite sure how your method works out.
     
  17. Nov 10, 2011 #16

    phinds

    User Avatar
    Gold Member
    2016 Award

    I don't know what the median has to do with anything and the CoM of a cone, if I have it right is at H/(1-SQRT(2)) assuming the cone is pointed up
     
  18. Nov 10, 2011 #17
    But the CoM of the cone is at H/4, assuming the cone is pointed up. There seems to be some miscommunication here. Edit: 1-sqrt(2) isn't even positive.
     
  19. Nov 10, 2011 #18

    phinds

    User Avatar
    Gold Member
    2016 Award

    Yeah, sorry about that, got rushed. Meant to say 1-(1/sqrt(2)), or in other words, .707.

    I'm running a "stack of disks" computer program just now but seem to have something messed up since instead of .707, or the .75 you say is the right answer, I'm getting .794

    Is the center of mass of a cone in a different place than the center of area of a triangle created by intersecting a plane through the axis of the cone? If it is, then I've got it wrong 'cause that's what I'm basing it on.
     
  20. Nov 11, 2011 #19
    Was your assumption that the CoM of the cone is vertically at the same location of the CoM of the triangle? If so then that's not right. When you rotate the triangle, more weight is prescribed to the larger end, so intuitively the CoM should move towards the larger end.
     
  21. Nov 11, 2011 #20
    The com of the section area as almost nothing to do with the com of the solid.
    For example consider two cubes: the first with volume 1 and the other with volume 8.
    The com of the section is at 1/5 the distance from the bigger one; the real com is at 1/9.

    If you write the integral you will see that the com of the cone is related to that of the 2d figure which has two parabolas as borders.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: How do I find the center of mass of a cone?
  1. How do I solve this? (Replies: 11)

  2. How do i find the area (Replies: 1)

  3. How do I write this? (Replies: 11)

Loading...