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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?
because I came across something called the Riemann Stieltjes integral and it is making no sense to me
This is a fairly common question:
https://www.physicsforums.com/showthread.php?t=95510
To determine the centroid of a general figure, you certainly need Calculus.
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?
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
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.
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.
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
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.
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.
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.