The motion of the center of mass in various binary systems can be described as follows: In binary star systems, two stars orbit around a common center of mass that is closer to the more massive star with their periodic motion influenced by gravitational attraction.
In diatomic molecules, the center of mass lies along the line connecting the two atoms, and its motion changes during vibrations, rotations, or reactions, affecting molecular interactions. In the Earth-Moon system, the center of mass is located inside the Earth, with both bodies orbiting this point, which influences tidal forces and their relative positions.

The center of mass can be found by using the concept of symmetry and geometric principles. For a rectangle, the diagonals cross each other at its center of mass. In the case of a circle or a cylinder, it is located at the middle of the geometric figure. In a solid sphere, it also lies at its center.
For a triangular plate, it lies at the point of intersection of the medians. In regular polygons, it is at the intersection of diagonals or at the centroid of the shape which is obtained by averaging the coordinates of the vertices.