Newton's Second Law for a System of Particles

For a single particle of mass m we already have encountered Newton's second law, namely , and  are all the forces acting on the mass m and  is the resulting acceleration of the mass m. What happens for a system of particles? The end result is

    (1.1)

where  is the sum of all external forces acting on the body (all the internal forces cancel out to zero), M is the total mass of the body and cm is the acceleration of the center of mass of the body.

Example Prove equation (1.1).

Solution Recall our definition of center of mass, namely

or

Taking the time derivative gives

and taking the time derivative again gives

which is just the sum of all the forces acting on each mass mi. These forces will be both external and internal. However for a rigid body all the internal forces must cancel because in a rigid body the particles don't move relative to each other. Thus just becomes  in agreement with (1.1).