Where:

We can also write Newton's second law for a system of particles:

Where:

Note that there is no restriction in the way the system of particles are connected. As a result, the above equation will also hold true for a rigid body, a deforming body, a liquid, or a gas system.

The above equation can be proven as follows.

Define the following, as shown in the figure below:

Let XYZ represent an inertial reference frame.

Let point

Let

Let

Let

Let

For each particle

If we sum this equation over every particle in the system we have

where

On the left side of the above equation all the internal forces cancel out (since they are equal and opposite). This leaves us with only the external forces acting on the system of particles. Thus we have,

The right side of this equation can be replaced with the total mass of all the particles in the system times the acceleration of the center of mass. Therefore,

This is the result we set out to prove.

Note that the center of mass of a system of particles can be a point in empty space. It does not need to be at the location of a particle.

See the page on center of mass to see how to determine the location of the center of mass of a system of particles.

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