Center of Mass (CM)

Dynamics of a particle system

OK so. The system we’re about to deal with.

When it’s made of a bunch of particles, and each of those particles is sitting at its own different position… yeah. Handling that directly is ridiculously messy.

So we introduce this thing called the center of mass (CM).

The idea: take all those particles, lump their masses together, and imagine that whole lump plonk — sitting at one position. That’s it.

First, the equation (it’s not a formula — it’s an equation):

$$\overrightarrow{r_{cm}} = \frac{\sum \overrightarrow{r}_{i} m_{i}}{\sum m_{i}}$$

Look at this for a sec. For starters — we’re dividing a vector by a scalar, so…

it’s a vector. Obviously!!!

Now look at the numerator!!!! (not a reactionary element… T_T T_T T_T sorry lol)

We take the position of the $i$-th particle,

$r_{i}$

and to that vector, we multiply the mass of that $i$-th particle,

$m_{i}$

— a scalar — and then sum them all up.

Hmm, so what this is saying:

If the little mass is kinda big, the vector gets exaggerated — bigger than the actual position vector.

If the little mass is less than 1, the vector gets shrunk — smaller than the actual position vector.

And you add aaall those scaled vectors together. That’s the numerator.

OK let’s suppose every particle has the same mass $M_i$.

Then

$$\sum r_{i} m_{i}$$

since the weights (the masses) multiplying each vector are all identical, the total summed vector would… lean toward wherever there are more particles, right?!!

Ah. OK. I’m getting it now.

We multiplied the scalar $M_i$ (the weight) onto the vector $r_i$ and summed. And if we then divide that whole sum by the total mass — boom, that’s an average!!!

So the center-of-mass vector — you can think of it as the average position of the particles, weighted by mass. Each particle gets a say, and its say is proportional to how heavy it is.

Now now now now — those particles move~~

The center-of-mass velocity vector, same deal: take each particle’s velocity vector, multiply by that particle’s mass $M_i$, divide by total mass.

Let’s see how this relates to the center-of-mass position vector.

$$v_{cm}=\frac{\sum m_{i}\dot{r_{i}}}{\sum m_{i}}=\frac{\sum m_{i}\frac{d}{dx}r_{i}}{\sum m_{i}}=\frac{\frac{d}{dx}\sum m_{i}r_{i}}{\sum m_{i}}=\frac{d}{dx}r_{cm}=\quad\dot{r_{cm}}$$

Differentiate the position vector of the center of mass with respect to time → you get the velocity vector of the center of mass.

Obvious? Yeah, obvious.

Alright, I’ll skip writing out the acceleration vector of the center of mass.

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