Q:

Center of Mass in 3D object?

How would I find the center of mass in a 3D object (a "spinning top" or "dreidel") that consists of a cylinder welded on top of a box welded on top of an upside down cone? Assume building material is all the same type
Here is a wireframe render of the object:

And here are the dimensions as created:
Top Cylinder: Radius=12, Height=50
Middle Box: Width=70, Height=56, Depth=70
Bottom Cone (inverted after creation): Radius: 35, Height: 28

A:

Use this facts:

COM of Cylinder and Cube are at body centre.
COM of Cone at height $h/4$ from base.

Now do weighted AM of coordinates of COM three figures with volume as their weights. 

Let's do Calculation.
Simple symmetry says COM will be along the line perpendicular to ground through the point from the cone touching ground. So let's do single calculation:
$$\frac{((\pi\times12^2\times50)\times(56+28+25))+((70\times56\times70)\times(28+28))+((\frac13\times\pi\times35^2\times28)\times(\frac34\times28))}{(\pi\times12^2\times50)+(70\times56\times70)+(\frac13\times\pi\times35^2\times28)}\approx 55.82$$
So height above ground must be nearly $55\sim56$ .

