How can we find the center of mass of a solid cone?

AI Thread Summary
To find the center of mass of a solid cone, a method involving nested hollow conical shells is proposed. The first step identifies the center of mass at y = 2H/3, where H is the height of the cone. The density is calculated as ρ = 3M/πR²H, leading to a discussion on how to determine the mass element "dm." It is clarified that using infinitesimal heights "dh" is inappropriate since the heights of the hollow cones are not infinitesimal. The conversation emphasizes the need to calculate the mass and center of mass for each hollow shell from the common base.
Nimarjeet Bajwa
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Homework Statement
Find the center of mass of a solid cone taking a hollow cone as an element. given that both coens are made of same material
Relevant Equations
-
is this method even possible? anyways here is my attempt

Step1) y= 2H/3 ( H is the height of the cone)

step 2) we take the density (ρ)= 3M/π R2 H.

The problem i am facing is to Find "dm"
 
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Nimarjeet Bajwa said:
Homework Statement: Find the center of mass of a solid cone taking a hollow cone as an element. given that both coens are made of same material
Homework Equations: -

is this method even possible?
Yes.
Consider a nested stack of hollow conical shells each of thickness dr. Find the mass and mass centre of each (measured from the common base).
 
haruspex said:
Yes.
Consider a nested stack of hollow conical shells each of thickness dr. Find the mass and mass centre of each (measured from the common base).
should we also take a height of every hollow cone as "dh" ?
 
Nimarjeet Bajwa said:
should we also take a height of every hollow cone as "dh" ?
The heights will not be infinitesimals, so dh would be inappropriate.
If the radius of one of these hollow cones is r, what is its height?
 

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