The average position of the mass in a body or system. A system will behave in response to external forces applied to any of its parts as if the entire mass of the body were concentrated there. The motion of the center of mass is unaffected by internal forces in the system (e.g. forces between the atoms, or collisions between different components of the system).
Motivation for Concept
While the motion of a point particle in response to well-defined external forces is relatively straightforward to determine, it is not so simple to calculate the motion of an extended body, especially if it is not rigid. The motion of the center of mass of a body, however, will act as if all of the mass of the body were concentrated there as a point particle, and will act the same way a point particle will in response to the same external forces. This is true even if the body is not rigid, if it is rotating, and regardless of where the forces are applied (e.g. even at the edge of the body, or to only one part of the system). This is therefore a very useful concept fot visualizing and calculating the dynamics of bodies that are not themselves point masses.
Mathematical Definition
Body as Sum of Point Particles
Under this condition, we can quickly derive the form and the utility of the moment of inertia by considering the body to be a collection of Np point particles. Each of the Np point particles (of mass mi where i runs from 1 to Np) will obey [Newton's 2nd Law|Newton's Second Law
Center of Mass as a Sum
We define the Center of Mass as:
Calculating Center of Mass
Integrals in Rectangular Coordinates
For continuous objects, the summation in our definition of the center of mass must be converted to an integral. The Mass differential is given by:
To calculate the x - position of the center of mass we use:
We can obtain the y - and z - coordinates using similar expressions:
