# CG of a cone ( using hollow cone)

The problem statement,
I know how to find the cg of a solid by using cross section
but i just don't know how to find the cg of the cone by using the cg of a hollow cone

for eg, we can calculate the cg of the half sphere by 1. calculating the cg of the hollow half sphere, than use it to calculate the cg of solid sphere

the problem is that i can't write a equation for the dm of each hollow cone because x and y changes

i just wanna know how to do it and in fact this is not a hw question

can anyone help me plz?

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I am assuming that the cone is right-circular, with an apex angle of t. Lets say that the cone is placed with its circular surface on the ground.
You can say that the solid cone is made of lots of hollow cones, one placed over the other.
Lets find the mass and cm of hollow-cone first, and then use integration to find cm of solid cone..
Hollow cone:
Hollow cone has its cm at a height of h/3(=r/3 *tant) from the bottom. Let the base of this hollow cone have inner radius 'r' and outer radius 'r+dr'.
Then, volume of hollow cone = d(volume of solid cone)
=d( (pi/3) r^3 tant) = pi*tant*r^2*dr. --> its mass = density*volume
Solid cone:
Therefore, cg of solid cone is
integral (r/3 *tant * pi *tant* r^2*dr*density)/integral(pi*tant*r^2*dr*density)
with the limits of r being 0 to R(base radius of solid cone).
You get the final answer as R*tant/4 which is same as H/4.

The horizontal cordinate of cm of hollow cone is zero... implies horizontal cordinate of cm of solid cone is also zero.

let me know if u understood this.. or need additional help