Center of Mass - Linear and Angular Momentum

Linear and Angular Momentum

The linear and angular momentum of a collection of particles can be simplified by measuring the position and velocity of the particles relative to the center of mass. Let the system of particles P_i, i=1,...,n be located at the coordinates r_i and velocities v_i. Select a reference point R and compute the relative position and velocity vectors,

The total linear and angular momentum vectors relative to the reference point R are

and

If R is chosen as the center of mass these equations simplify to

For any system with no external forces, the center of mass moves with constant velocity. This applies for all systems with classical internal forces, including magnetic fields, electric fields, chemical reactions, and so on. More formally, this is true for any internal forces that satisfy Newton's Third Law.

The total momentum for any system of particles is given by

where M indicates the total mass, and v_cm is the velocity of the center of mass. This velocity can be computed by taking the time derivative of the position of the center of mass. An analogue to Newton's Second Law is

where F indicates the sum of all external forces on the system, and a_cm indicates the acceleration of the center of mass. It is this principle that gives precise expression to the intuitive notion that the system as a whole behaves like a mass of M placed at R.

The angular momentum vector for a system is equal to the angular momentum of all the particles around the center of mass, plus the angular momentum of the center of mass, as if it were a single particle of mass :

This is a corollary of the parallel axis theorem.