9.13:
Center of Mass: Introduction
Consider a point on the edge of a spinning disc following a parabolic path during a discus throw event. The point takes a complex path even when the disc continues to move in a parabolic path, whereas the center of the disc follows the same path as that of the disc.
This point is called the center of mass rcm of the disc. The concept of center of mass helps to illustrate the object as a point mass. During the motion, the object's center of mass moves such that the total mass is condensed in a single point.
Consider several particles of masses mi with respective position vectors ri. Then, the position vector for the center of mass is written as the summation of mass times position vector divided by the total mass of all the particles.
The coordinates of the center of mass can be expressed as the summation of the product of mass and the individual coordinates of the position vector divided by the total mass.
9.13:
Center of Mass: Introduction
Any object that obeys Newton's second law of motion is made up of a large number of infinitesimally small particles. Objects in motion can be as simple as atoms or as complex as gymnasts performing in the Olympics. The motion of such objects is described about a point called the center of mass of the object. The center of mass of an object is a point that acts as if the whole mass is concentrated at that point. The center of mass of an object with a large number of infinitesimally small particles is calculated by taking the product of each particle's mass and its position with respect to the defined origin and dividing it by the total mass of the object.
The center of mass of an object does not necessarily have to be a physical point inside the object. For example, in a sphere, the center of mass is at the center of the sphere; however, the center of mass of a doughnut is at the center, where there is no mass. The center of mass for any symmetrical object lies on the axis of symmetry and the plane of symmetry. For any extended object, if the acceleration due to gravity is constant over the whole mass, then the center of gravity and the center of mass will be identical for that object.
This text is adapted from Openstax, University Physics Volume 1, Section 9.6: Center of Mass.