What you Need
What to do
Point the index fingers of both hands and hold your hands apart but parrallel.
Get someone to balance the broom horizontally on your two index fingers, making sure you keep them level.
Now gently move your fingers together.
What may happen
You should find that as long as you keep your fingers level the broom stays balanced, even as they get very close together. You have found the broom's centre of gravity.
Why does it happen?
When you start off holding the broom inevitably one finger will be closer to the centre of mass than the other.
This means that more of the weight will be sitting on the closer finger, so it must be pushing upwards harder.
The larger the force between the finger and the broom the larger the maximum frictional force it can provide before it slides.
The finger furthest from the centre of mass will therefore be able to slide the most easily, and it moves until it is no longer the furthest from the centre of mass.
Then the other finger can move, so they keep sliding in such a way as they move closer and closer to the centre of mass and the broom stays balanced.