Center of mass of a Frustrum cone???

Question

What is the center of mass of a Frustrum cone in the y direction?

Answer 1

I assume you are asking for the center of mass of the frustum of a right circular cone. (The math gets veryy messy if the cone is not a right cone, or if it is not circular.) I also assume the frustum has a uniform density throughout.



In accord with your question, let the frustum have its axis along the y-axis, and it's base on the x-z plane. Let the radius of the base be R, the height of the cone from which the frustum is cut be H, the height of the frustum be h (h< H, obviously), and D be the density of the material.



The radius of the frustum as a function of distance along the y axis is given by:



r(y) = R- y*R/H = (H-y)*R/H, 0 <= y <= H



The mass loading (the linear mass density) as a function of distance along the y-axis is given by:



M(y) = D*pi*(r(y))^2 = D * pi * (H-y)^2 * (R/H)^2



The total mass between two values of y is given by the integral of M(y) dy between the two points. We want to find a point y = C such that:



Integral from 0 to C of {D * pi * (H-y)^2 * (R/H)^2 dy} = Integral from C to h of {D * pi * (H-y)^2 * (R/H)^2 dy}



That is, essentially want to find the value y = C for which the volume to one side of C is equal to the volume on the other side of C (we can use volume instead of mass because we've assumed that the density is uniform).



Cancelling like factors on either side of the above equation, we have that:



Integral from 0 to C of {(H-y)^2 dy} = Integral from C to h of {(H-y)^2 dy}



-1/3 * (H-C)^3 + 1/3 * H^3 = -1/3 * (H-h)^3 + 1/3 * (H-C)^3



H^3 - 2*(H-C)^3 + (H-h)^3 = 0



This has only one real root:



C =H + 0.5*(12*h*H^2 + 4*h^3 -12*H*h^2 - 8*H^3)^(1/3)



The center of mass of a right circular frustum with uniform density and with its axis along the y axis is at x = 0, y = C, z = 0, where C is given above.

Answer 2

First, I wouldn't trust the above poster's math and/or reasoning even though the logic looks impressive at a first glance. The logic about solving for something with a root is troubling because center of mass calculations are relatively straightforward as long as the math is performed properly. There ought not be any taking roots or eliminating solutions....



You can start here:

http://www.efunda.com/math/solids/CenterOfMass.cfm

http://www.efunda.com/math/solids/solids_display.cfm?SolidName=RightCircularCone



You may have to refer to an engineering handbook for the exact formula relating to the frustum. You could derive it by slicing some top portion of a right cylindrical cone and recalculate the related integrals and the like.



I also find it troubling that this forum appears to cache answers because I see a question with no answers, I prepare my answer and I find another answer - I've had this happen to me a number of times already. Maybe it's my bad luck or the inconsistencies of this forum....

Answer 3

it relies upon upon how thick is the wall of your hollow cone. Or did you recommend some perfect mathematical cone, with an infinitely skinny floor, which by some ability could nevertheless weigh something? besides, with out information on the thickness, won't be able to calculate in theory those issues are ordinary to compute, integrating over the quantity or floor the infinitesimal mass aspects dm, situations the distances to a chosen reference element. In practice although, the algebra can replace right into somewhat gruesome ;-)