Summarised and decomposed quantities

The centre of mass is not just a geometric point: it is a genuine imaginary body in which some quantities of the system can be summarised, as if the whole mass were concentrated there. Other quantities, instead, split into two parts: a “CM” contribution and an “internal” one, relative to the CM.

Principle — Quantities summarised in the CM

The centre of mass has:

1. mass $M_{\text{CM}} = M_{\text{tot}}$, equal to the system’s total mass;
2. momentum $\vec{p}_{\text{CM}} = \vec{p}_{\text{tot}}$, equal to the total momentum;
3. gravitational potential energy $E_{\text{pot,g,CM}} = M_{\text{tot}}\,g\,h_{\text{CM}}$, equal to the total $E_\text{pot,g}$.

These three quantities are summarised totally: the CM carries all of them, and nothing “internal” is left over.

Other quantities — kinetic energy and angular momentum — are not fully summarised, but decompose into two contributions:

• the “CM” contribution, computed by treating the CM as a point mass $M_\text{tot}$;
• the internal contribution, computed in the CM frame, where velocities are relative to the CM:

$E_\text{cin,tot} = E_\text{cin,CM} + E_\text{cin,int} \qquad \vec{L}_\text{tot} = \vec{L}_\text{CM} + \vec{L}_\text{int}$

Summary diagram of the CM’s quantities. Mass, momentum and $E_\text{pot,g}$ are fully summarised in the CM. Kinetic energy and angular momentum instead have both a “CM” term and an “internal” one.

Links

Topics: Centre of mass Concepts: Centre of mass · Momentum · Kinetic energy · Angular momentum

Related exercises: Velocity and momentum of the CM · Finding the masses from the CM · Kick and impulse