Center of Mass must Lie on the plate?

Question

My teacher told me that the following statement is false:

The center of mass of a thin plate must lie on the plate.

How do I prove that it is so? (We can use an explanation or a counterexample)

I was thinking that since x̄ $=\frac{M}{m}$, if the mass is equal to 0, then the plate's center of mass could be off the plate? Another thing I was thinking about was a donut shape, but that wouldn't be the definition of a plate, would it?

Thanks for any clarification or help!

Answer 1

Hint: Think about a flat plate shaped like a kidney; or like a right angle.

In general, if the shape is convex then the center of mass will lie on the plate. If it is not convex, then then the center of mass can be off the plate.