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?