Centre of mass — definition for system of particles: Foundations

This is the toolbox page for the parent topic. We will pick up each tool the parent uses, hold it up, say what it means, draw the picture it stands for, and explain why the topic can't proceed without it. Nothing here assumes you have seen the notation before.

0. The scene we are describing

Before any symbol, picture the physical situation the whole chapter is about.

We have several small blobs of matter (call them particles) floating in space. Each one is at some location and each one has some heaviness. That is the entire raw material. Every symbol below is just a compact name for one piece of this picture.

1. A "particle" and its label $i$

Picture: in the figure above, each blob wears a numbered tag. $i$ is nothing more than that tag.

Why the topic needs it: we have many particles and must talk about "the mass of that specific one". Without a label we could not tell them apart. The subscript in $m_{i}$ is exactly this tag glued onto a symbol.

2. Mass $m_{i}$ and total mass $M$

The capital $M$ is built from the small $m_{i}$ by summing (next section): $M = m_{1} + m_{2} + m_{3} + \dots$

Picture: the size of each blob in the figure hints at its $m_{i}$ . Melt all blobs into one lump — that lump's mass is $M$ .

Why the topic needs it: the centre of mass is a mass-weighted average, so the individual heavinesses $m_{i}$ decide how much each particle "pulls". Dividing by $M$ at the end is what turns a raw total back into a proper average.

3. The summation sign $\sum$

This is the single most important piece of notation to nail.

The letter $n$ here is just "how many particles there are in total".

Why this tool and not writing out the pluses? Because a system might have $3$ particles or $3000$ . The $\sum$ symbol writes one formula that works for any count. It answers the question: "how do I express 'add over the whole crowd' without knowing the crowd size?"

4. Position, coordinates, and the vector $r_{i}$

To say where a particle is, we need a fixed reference point and a way to measure.

Why the topic needs vectors: particles live in 2D or 3D, so "position" needs a direction, not just a number. The vector $r_{i}$ packs all three coordinates into one symbol. That is why the parent's master formula $r_{c m} = \frac{1}{M} \sum_{i} m_{i} r_{i}$ can be written on one line — and then unpacked into three separate coordinate equations for $x$ , $y$ , $z$ .

5. Weighted average — the heart of the formula

The COM is literally a weighted average of positions, so this idea must be crystal clear.

For the COM, the weights are the masses and the values are the positions. See Weighted Average and Moments for the same maths in the language of levers.

Why this tool? A plain average would treat a feather and a boulder equally — wrong, because the boulder should dominate "where the mass is". The weighted average is the only averaging rule that lets heavier particles pull the result toward themselves, which is precisely the physical behaviour we want.

6. Rate of change: velocity, and the symbol $\frac{d}{d t}$

The parent's derivation (Section 2) needs one more tool: how fast a position changes.

Picture: freeze two snapshots of a moving blob a whisker of time apart; the tiny arrow from the first spot to the second, divided by that tiny time, points along $v_{i}$ and has length = speed.

Why this tool and not just "distance ÷ time"? Because velocity can change moment to moment. The derivative $\frac{d}{d t}$ measures the instantaneous rate — the value right now, not averaged over a long trip. The parent uses it to define momentum $P = \sum_{i} m_{i} v_{i}$ and then to demand $P = M v_{c m}$ , which is what forces the COM formula into existence.

Recall What does

$\frac{d r _{i}}{d t}$ give you? The instantaneous velocity $v_{i}$ of particle $i$ — its position's rate of change. ::: velocity

7. Putting the symbols together

Now the parent's headline formula is fully readable: $r_{c m} = \frac{1}{M} \sum_{i} m_{i} r_{i}$

Read it aloud with everything we built:

Every piece is a tool from a section above — nothing is left unexplained.

Prerequisite map

Equipment checklist

Test yourself — answer each before revealing.

What does the subscript

i

m_{i}

mean?

A label/tag pointing at "the

i

-th particle"; it is never multiplied.

What operation does

\sum_{i}

perform?

It adds a quantity over every particle in the system.

What is the difference between

m_{i}

and

M

m_{i}

is one particle's mass;

M = \sum_{i} m_{i}

is the total of all masses.

What does the arrow in

r_{i}

signify, and what are its pieces?

It marks a vector (length + direction); its pieces are the coordinates

(x_{i}, y_{i}, z_{i})

measured from the origin.

In a weighted average, what plays the role of the weights for the COM?

The masses

m_{i}

— heavier particles get more pull.

Why must you divide the sum

\sum_{i} m_{i} x_{i}

M

The sum has units kg·m (a moment); dividing by

M

(kg) gives a position in metres.

What does

\frac{d r _{i}}{d t}

represent physically?

The instantaneous velocity

v_{i}

of particle

i

Write the single-line vector formula for the centre of mass.

r_{c m} = \frac{1}{M} \sum_{i} m_{i} r_{i}

Connections

Parent: COM definition — where every tool here is used.
Weighted Average and Moments — the pure-maths engine behind Section 5.
Newton's Second Law for a System of Particles — needs the derivative and momentum tools of Section 6.
Conservation of Linear Momentum — builds on $P = M v_{c m}$ .
Centre of Mass of Continuous Bodies — replaces $\sum$ by $\int$ once summation is understood.
Collisions — Elastic and Inelastic — analysed using these foundations in the COM frame.