CENTER OF MASS of a circular conical surface

[Meister RaRo]

Homework Statement

Find the center of mass of a circular conical surface (empty cone) of height H.

Homework Equations

z(CM) = 1/M * int(z*dm)
x(CM)=y(CM)=0 (we've taken the origin of coordinates at the center of the base)

The Attempt at a Solution

This problem far exceeds my mathematical knowledge so I'll probably be talking nonsense but... here it goes:
let L be the slant height of the cone, R the radius of the circular base
dm= (M/pi*R*L)*dS
dS= 2pi*r*dz (where r is the "radius of each dS", r= R(H-z)/H)
We can now solve the integral, obtaining:
z(CM)= H^2/3L
But, given the problem wording, z(CM) should depend only upon H...!

Thank you for your help (and excuse my lousy English)!

 

[ozymandias]

Unless I've misunderstood your question, the result can definitely not depend on H alone.
Consider two cones of height H, one with a base of radius R1 and another with a base of radius R2.
Now take R1>>R2. It seems clear to me that cone #1 will have its center of mass well below that of cone #2.--------
Assaf
http://www.physicallyincorrect.com/"

 

[ogb p]

Your approach to solving this problem is correct. The center of mass (CM) of any object is the point where the object can be balanced without any rotation. In this case, the center of mass of the circular conical surface will lie on the axis of symmetry, which is also the axis of rotation.

To find the center of mass, we use the equation z(CM) = 1/M * int(z*dm), where z is the distance from the origin (center of base) to the infinitesimal mass element dm. In this case, we can use cylindrical coordinates to represent the infinitesimal mass element as dm = (M/πR^2) * rdrdθ, where r is the distance from the axis of symmetry and θ is the angle measured from the axis of symmetry.

We can then integrate over the entire surface to find the center of mass, using the limits of integration as θ = 0 to 2π and r = 0 to L (the slant height). This leads to the expression z(CM) = H^2/3L, which is the same result you obtained.

Therefore, the center of mass of a circular conical surface lies at a height of H^2/3L from the center of the base, regardless of the dimensions of the cone. This is because the shape of the cone remains the same as long as the ratio of the height to the radius remains constant.

I hope this helps in your understanding of the concept of center of mass. Keep up the good work in your scientific pursuits!