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My Question: For finding the Center of Mass ($COM$) of a hollow cone, why do we use its area to define its elemental mass ($dm$) and not its volume, which we use to find the $COM$ of a solid cone.

The formula I use for finding the center of mass of an object is:

$$\frac{\int_{o}^{h}dmy}{\int_{o}^{h}dm}$$

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    $\begingroup$ For a solid cone, the mass is dependent on volume , for a hollow, it is dependent on the surface area. $\endgroup$
    – Linkin
    Commented Oct 6, 2020 at 12:47
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    $\begingroup$ The hollow cone does not have a 'volume,' at least in the theoretical sense. $\endgroup$
    – Yejus
    Commented Oct 6, 2020 at 12:55

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Your mass element for a hollow cone will be dm = ρ(2πr)dl where ρ is the surface mass density, dl is a segment of length on the surface measured from the apex, and r is measured from the central axis to dl. If y is measured along the axis from the apex, then r = y tan(θ) and dy = dl cos(θ).

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