This lab experiment is based on the fact that when an object is allowed to pivot about a point, the force of gravity will be exerted through the center of gravity of the object so as to rotate the object such that the center of gravity will always be vertically aligned within the pivot. Only when this alignment is complete does the moment due to gravity about the pivot become zero ($0$).
Center of Gravity Lab
How to find the center of mass of a lamina (a leaf)