Introduction
This is a little more complicated than just center of mass, but it is ultimately center of mass that keeps it from falling over. The center of mass is at the center where the lower piece of wood touches the nail that's sticking out of the center of the base that really doesn't effect the way the nails balance. This is actually a matter of the torque from both sides balancing out, creating an equilibrium.
Principle
This demonstration is made possible by torque, which is calculated by multiplying force and the distance from the center of mass.
Materials
9 nails
a large wooden base
2 smaller pieces of wood
2 hair ties