Theorem on the Motion of the Center of Mass

The theorem on the motion of the center of mass describes the translational motion of a mechanical system by a single equation in which internal forces do not appear explicitly.

Derivation

For each particle, $m_i\vec a_i=\vec F_i^{e}+\vec F_i^{i}$. Summing over the system cancels the internal forces, while $\sum m_i\vec a_i=M\vec a_C$.

Fundamental equation

Therefore:

$$M\vec a_C=\sum\vec F^{e}=\vec R^{e},$$

where $\vec R^{e}$ is the resultant of the external forces.

Coordinate components

In Cartesian coordinates:

$$M\ddot x_C=\sum F_x^{e},\qquad M\ddot y_C=\sum F_y^{e},\qquad M\ddot z_C=\sum F_z^{e}.$$

These equations have the same form as the equations of motion of a particle of mass $M$.

Zero external resultant

If $\sum\vec F^{e}=0$, then $\vec a_C=0$ and $\vec v_C=const$. The center of mass is either at rest or moves uniformly in a straight line.

Conservation along one coordinate

If the sum of external-force components along one axis, say $x$, is zero, then $v_{Cx}=const$. If additionally $v_{Cx}(0)=0$, the coordinate $x_C$ remains constant.

Internal motions

Internal forces can strongly change the relative positions of system parts, but by themselves cannot change the center-of-mass motion of an isolated system. Motion of one part is therefore accompanied by compensating motion of other parts.

Example

A horizontal external resultant of 30 N acts on a system of mass 10 kg. Regardless of internal interactions, $a_C=30/10=3$ m/s² in the force direction.

Common mistakes

• including internal forces in the center-of-mass equation after summing the complete system;
• assuming $\sum\vec F^e=0$ means the center of mass must be stationary rather than have constant velocity;
• applying the theorem to only part of a system without reclassifying forces;
• identifying center-of-mass motion with the motion of every system particle.