Dynamics of a particle system

So suppose the system we’re dealing with…

…is made of many particles, each at its own position.

Handling all of them individually is painfully complicated — so we bring in the concept of the center of mass (CM).

Rough idea: lump all the particles together and pretend the total mass sits at a single point.

Let’s start with the formula (it’s a definition, not an axiom):

$$ \vec{r}_{\mathrm{cm}} = \frac{\sum \vec{r}_i \, m_i}{\sum m_i} $$

Stare at it for a moment — a vector divided by a scalar, so the result is still a vector. Good.

Now look at the numerator.

The position of the $i$-th particle,

$$\vec{r}_i$$

is multiplied by that particle’s mass,

$$m_i$$

and we sum them all.

So:

  • A particle with larger mass gets its position vector amplified in the sum.
  • A particle with mass less than one (in whatever units) gets its position vector shrunk.

Then all those weighted vectors are added together — that’s the numerator.

Imagine every $m_i$ is the same. Then

$$\sum \vec{r}_i m_i$$

is a uniformly-weighted sum of position vectors — which will “lean toward” wherever the particles are clustered. Makes sense.

So we’re taking each position vector $\vec{r}_i$, scaling it by the scalar weight $m_i$, and summing. Dividing by the total mass gives us a weighted average.

In other words, the CM vector is just the mass-weighted average position of the particles.

Now let the particles move.

By the same logic, the CM velocity should be the mass-weighted average of each particle’s velocity, divided by the total mass. Let’s see what that actually equals:

$$ \vec{v}_{\mathrm{cm}} = \frac{\sum m_i \dot{\vec{r}}_i}{\sum m_i} = \frac{\sum m_i \frac{d}{dt}\vec{r}_i}{\sum m_i} = \frac{\frac{d}{dt}\sum m_i \vec{r}_i}{\sum m_i} = \frac{d}{dt}\vec{r}_{\mathrm{cm}} = \dot{\vec{r}}_{\mathrm{cm}} $$

The time derivative of the CM position vector is exactly the CM velocity vector. Obvious? Yeah, obvious. But it’s nice to see it fall out cleanly.

CM acceleration follows the same pattern — I’ll skip writing that one out.

Originally written in Korean on my Naver blog (Nov 2014). Translated to English for gdpark.blog.