This activity makes me happy because it reminds me of the one activity I distinctly remember from high school physics with Mr. Eide, master of the pun. We used large pieces of Styrofoam and a soldering iron (I think) to make the holes. This version is safe to do with young students and the materials are easier to come by.
You'll need to make a plumb line (or several).
Cut a length of string, about 12 inches. Tie a metal washer to the end (or a fishing sinker or anything else that's heavy and can easily be tied to the end of a string).
Tie the other end of the string around a pushpin.
Cut one side of a manila folder into an irregular shape
Punch five holes (randomly spaced) at the edge of the shape.
Stick the pushpin (with the plumb line attached) through one of the holes and hang it on a bulletin board. The shape and the string should both swing freely.
When everything has stopped moving, use a pen to draw a line on the paper along the string.
Move the pushpin to another hole and repeat.
Continue until you've used each of the 5 holes.
Take the shape down. The five lines intersect in one point. If you place your finger under that point, the shape will balance perfectly. You've located the center of gravity for your shape!
And, if you move your finger, even a small amount, from that spot, the shape will tumble to one side or the other!