Momentum of a System of Particles

In mechanics, the thing that works like a particle’s (or object’s) ID card is momentum. That’s the word.

But we’re not dealing with one object anymore — we’ve got a whole system of particles. So we redefine:

The momentum of the particle system is the vector sum of the momenta of each particle!!!

$$\mathbf{P} = \mathbf{P}_1 + \mathbf{P}_2 + \mathbf{P}_3 + \cdots + \mathbf{P}_n$$

(Bold = vector. Vectors! Going forward, always!!)

For each particle,

$$p_{i} = m_{i}v_{i}$$

so

$$p = \sum m_{i}v_{i}$$

Huh — this looks familiar. Let’s multiply and divide by the total mass:

$$p = \frac{\sum m_{i}v_{i}}{\sum m_{i}}\sum m_{i} = v_{cm} \cdot \sum m_{i} = m \cdot v_{cm}$$

Ohhh. So the total momentum is just total mass times the velocity of the center of mass. Clean. Moving on.

Now let’s look at force ($F$).

The force on particle $i$ is

$$m_{i}a_{i} = m_{i}\ddot{r}_{i}$$

right.

What kinds of forces hit each particle? Two flavors, broadly:

1. Forces from outside the system.
2. Forces from inside the system — the interactions between particle $i$ and the other particles.

So each particle is getting pushed by both kinds at once. Which gives us:

$$m_{i}\ddot{r}_{i} = F_{i} + \sum_{j} F_{ij} = \dot{p}_{i}$$

And summing over the whole system:

$$\sum \dot{p}_{i} = \sum_{i} F_{i} + \sum_{i}\sum_{j} F_{ij}$$

Now look at that double-sum on the right. For every $F_{ij}$ there’s an $F_{ji}$ pointing the opposite way — Newton’s third law, baby — so pair by pair they cancel. Add them all up, you get zero!!!!

Think about it physically: we’re zoomed out on the whole system. Internal forces are particles shoving each other around inside, so from the system’s point of view they wash out. Double sum → 0.

That leaves:

$$\sum F_{i} = m a_{cm}$$

right. Heh. OK now — what if there’s no external force at all? Whoa. Then $m a_{cm}$ is $0$. Amazing. So $m \dot{v}_{cm} = 0$?? Which means $m v_{cm}$ has to be a constant (because the thing you differentiate to get zero is a constant).

But wait — $m v_{cm}$ is the total momentum we just saw!

gasp

Total momentum is constant??

I feel like I’ve heard this somewhere. A lot.

This is exactly the law of conservation of linear momentum!!!

And that’s us — all the way up to conservation of linear momentum (linear moment).

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Originally written in Korean on my Naver blog (2014-11). Translated to English for gdpark.blog.