An arbitrarily shaped piece of plywood can be suspended from a string attached to the ceiling. How could you use a plumb bob to find its center of mass?
Use the plumb bob to create two lines from two different points of suspension. The center of gravity will be the point where the two lines intersect.
If you suspend the sheet of plywood from the ceiling and let the plumb bob drop from the point of suspension, you can create a line that divides the sheet into two sides of equal weight.
Repeating this process from a second point of suspension will give you the center of gravity.