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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?
schaafde said:because I came across something called the Riemann Stieltjes integral and it is making no sense to me
HallsofIvy said:This is a fairly common question:
https://www.physicsforums.com/showthread.php?t=95510
HallsofIvy said:To determine the centroid of a general figure, you certainly need Calculus.
Guffel said: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?
phinds said: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
phinds said: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.
Yuqing said: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.
phinds said: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
Yuqing said: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.
phinds said: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.
phinds said: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.
The center of mass of a cone is the point at which the entire weight of the cone can be considered to act, making it the balance point of the cone.
The formula for finding the center of mass of a cone is (1/4) x (height of the cone) x (radius of the base squared) / (height of the cone) + (1/2) x (height of the cone). This can also be written as (3 x height of the cone) / (4 x height of the cone) + (1/2) x (height of the cone).
Yes, the center of mass of a cone will always be located on the central axis, as it is the point at which the weight of the cone is evenly distributed.
No, the center of mass of a cone will always be located within the boundaries of the cone itself.
The shape and dimensions of a cone will affect the location of its center of mass. A taller and narrower cone will have a higher center of mass, while a shorter and wider cone will have a lower center of mass. Additionally, a cone with a more pointed tip will have a higher center of mass than a cone with a flatter tip.