Center of mass of the three pennies?

Question

Three pennies each of radius R and mass M attached at their edges. How to find the center of mass of the three pennies?

Answer 1

Draw the equilateral triangle $ABC$ with corners the centres of the circles. Let $O$ be the centre of mass.

Now there are various ways to proceed. For example, drop a perpendicular from $O$ to $AB$, meeting $AB$ at $X$. Note that $AX=R$. Then $\frac{R}{OA}=\cos(30^\circ)=\frac{\sqrt{3}}{2}$. It follows that $OA=\frac{2R}{\sqrt{3}}$.

Answer 2

By symmetry, the center of mass is the barycenter of the triangle with vertex on the centers of the three pennies.