Where does the center of gravity of a fine rod 𝐴𝐵 of uniform density lie?
If we draw a diagram of this rod and divide it up by equally sized segments along its length, if we draw a vertical line through the center of the rod and start to eliminate segments of the rod on either side in pairs, we eventually get to a point where only two segments remain. And they’re equally positioned on either side of the center of this rod.
This means that the point at the very center of the rod through which we’ve drawn our line is at the rod’s midpoint, which is also its center of gravity. So halfway from one end of the uniformly dense rod to the other is the rod’s midpoint, which is also its center of gravity.