Centre of mass of the UK

 To find the centre of mass of an irregularly shaped object, you hang it so that it can swing freely and let it settle under gravity. There is then a plumb line hanging down from the suspension point. Draw a line vertically down the plumb line. Repeat for a different suspension point. Where the lines cross, that is the centre of mass. So the centre of mass lies directly below the suspension point.

 The reason is shown below. If I pull the object to the left side and out of equilibrium, there is more weight on the left of the plumb line that on the right. Each of the two divided sections of weight are act as a turning force (a moment). The moment on the left is an anti-clockwise moment, The moment on the right is a clockwise moment. But the anti-clockwise moment is bigger so there is a resultant moment pulling the shape back down. There is no resultant moment when equal weights are on either side of the plumb line. That can only happen when the centre of mass is below the suspension point.

If you suspend the object through the centre of mass, there will always be equal weights on either side of the middle. You can swing it to any position and it will stay there. There will never be a resultant moment.