Why are the volumes above and below the center of mass of a uniform cone not the same?

Question

So today in my physics class we derived the center of mass of a uniform cone, and it all made sense, but near the end of class a student asked,

"If you were to split an object into two parts with a plane through it's center of mass, would the two objects' masses be equal?"

And it got me thinking about our cone example in class. So I went and calculated the volumes above and below the center of mass of a cone to test the most basic case to see if my intuition held up.

It turned out that the volumes above and below the center of mass are not equal. My logic here was that since the object is uniform, the volume correlates to the mass, so the masses above and below are not equal. But if the masses above and below are not equal how can the center of mass be there? Can somebody explain to me why this is the case, or give me the calculations to prove that the volumes above and below the center of mass of a cone are equal?

Answer 1

You have a conjecture:

Cutting an object into two pieces using a plane through its center of mass yields two pieces with equal mass.

You have found a counter-example to this, so it must be false in general.

The reason this does not work in your case is because if each each piece had the same mass, then the COM of each individual piece would have to be equidistant from the COM of the original cone. This is not the case if we use a plane parallel to the base of the cone.

This shows the more general idea of center of mass: it depends on both the amount and the location (or distribution) of masses. For an even more simplified example, imagine two point masses with mass $m$ and $2m$ separated by some distance $d$. It is easy to show that the center of mass lies at a distance $\frac23d$ from the particle of mass $m$. Therefore, if we "cut" our system into two pieces at the center of mass, we would find that we still have particles of unequal mass.

Your problem is analogous to my simpler example. You can think of each new section after the cut as their own point mass located at their own center of mass. Each piece will have a different mass, and each center of mass will have a different distance to the original center of mass.

Of course this doesn't mean that there is never a way to cut the object through the COM and get two pieces with equal mass. All our reasoning shows is that it is possible for this not to be the case. The work that you claim to have done shows that this possibility actually occurs in reality.

Answer 2

Because mass that's further from the centre has more rotational inertia than mass closer. If you kick a can in its CM it moves without flipping end over end, if the can had more mass at the bottom its CM is lower. So it's not just the amount but the distance that's important.