Centroidally Bordered Bodies and the Balancing L

Abstract

The preparation of bodies which have the property that the mass center lies on the periphery is here considered. It is demonstrated that, starting from a large variety of shapes, including all those here called “medially convex,” it is always possible to prepare a centroidally bordered shape by cutting a simple triangular notch out of the basic shape. Incidentally it is proved that every area which is medially convex with respect to a base b is at least 8/3 times as great as the area of the centroid triangle constructed on the base b. The method of triangular notching is applied in detail to several familiar shapes, including the general trapezoid and circular segment. The preparation of a centroidally bordered shape from a basic shape which includes a straight angle in its boundary is also considered. The solution by quadrilateral notching is indicated but not carried through in detail. The simple solution for the special case of the balancing L is written down and illustrated photographically.