Foundations — Motion of centre of mass — external force determines a_CM

Before you can read ${\vec{F}}_{ext}=M{\vec{a}}_{CM}$, you must be able to read every mark in it. This page builds each symbol from nothing, in the order they depend on each other. If a mark shows up on the parent page, it is defined here first.

0 — What a symbol with an arrow on top means

The very first thing the parent page assumes is the little arrow: $\vec{r}$, $\vec{v}$, $\vec{F}$.

The picture. Stand at a chosen origin point $O$. To describe where something is, you don't just say "5 metres" — you say "5 metres, that way." That is an arrow from $O$ to the object.

Why the topic needs it. Positions, velocities, and forces all have directions. A skater pushing left is completely different from pushing right. A plain number (called a scalar, no arrow) could never tell left from right. So every core symbol on the parent page wears an arrow.

1 — $m_i$ : the mass of each piece, and the index $i$

The picture. Imagine numbered marbles: marble 1, marble 2, marble 3. Each has its own mass. The index $i$ is the sticker we put on each marble so we can talk about them one at a time without new letters.

Why the topic needs it. The system is many particles. We need one flexible symbol, $m_i$, to stand for "any of them," so that a single formula covers a system of 2 or 2000 pieces.

2 — ${\sum }_i$ : the summation sign (add up over all tags)

The picture. A basket you drop one term into for each particle, then read off the total.

Why the topic needs it. $M={\sum }_i{m}_i$ (total mass) and ${\vec{R}}_{CM}=\frac{1}{M}{\sum }_i{m}_i{\vec{r}}_i$ (the balance point) are both "do this for every particle and add." Without $\sum$ we'd have to rewrite the formula for each new number of particles.

3 — ${\vec{r}}_i$ : position, and ${\vec{R}}_{CM}$ : the balance point

Now we can build the star of the show.

The picture — why weighting by mass. Put a light ball and a heavy ball on a seesaw. The balance point is not the midpoint; it sits close to the heavy ball. Mass-weighting bakes exactly this in.

Why the topic needs it. This one point is what the whole topic tracks. The claim "internal forces can't move the CM" is a claim about how this specific average behaves.

4 — ${\vec{v}}_i$, ${\vec{a}}_i$ : velocity and acceleration (rates of change)

The parent page "differentiates" the position formula. Here is what that word means, with no calculus jargon.

Why the tool, and why not another? We want to know how the balance point moves and speeds up. Movement is change of position over time; a rate of change is precisely the tool that measures "per unit time." That is why the parent takes rates of change twice — once to reach velocity, once more to reach acceleration.

5 — ${\vec{p}}_i$ and ${\vec{P}}_{total}$ : momentum

The picture. A slow truck and a fast bike can carry the same momentum; a truck at rest carries none. The arrow's length is the oomph, its direction the travel direction.

Why the topic needs it. The gem ${\vec{P}}_{total}=M{\vec{V}}_{CM}$ (parent Step 2) links the crowd's total oomph to the single balance point's motion — the bridge to conservation of momentum.

6 — ${\vec{F}}_i$, and the internal/external split

Why the topic needs it. The entire result hinges on separating these two. Internal forces will cancel; external forces will survive to drive the CM.

7 — Newton's Third Law : why the internal pile cancels

The picture. Two skaters shove: each feels a push away from the other, equal in size, opposite in direction. Add the pair and you get the zero arrow — they annihilate.

Why the topic needs it. In ${\sum }_i{\sum }_{j\ne i}{\vec{f}}_{ij}$, every internal force appears alongside its equal-and-opposite twin. Summed, each twin-pair is ${\vec{f}}_{ij}+{\vec{f}}_{ji}=\vec{0}$. The whole internal pile vanishes — no matter how huge the forces are — leaving only external forces. See Newton's Third Law.

8 — $M$, ${\vec{V}}_{CM}$, ${\vec{a}}_{CM}$ : the whole-system symbols

Why the topic needs it. These are exactly the symbols in the master result ${\vec{F}}_{ext}=M{\vec{a}}_{CM}$ and $\vec{P}=M{\vec{V}}_{CM}$. Now every mark in those boxed formulas has a plain-words meaning and a picture behind it.

How these foundations feed the topic

Equipment checklist

Test yourself — cover the right side, answer, then reveal.

Connections

• Parent topic (Hinglish) — the page these foundations unlock.
• Newton's Third Law — the reason internal forces cancel in pairs.
• Conservation of Linear Momentum — what $\vec{P}=M{\vec{V}}_{CM}$ becomes when ${\vec{F}}_{ext}=0$.
• Collisions (Elastic & Inelastic) — momentum and CM ideas in action.
• Rocket Propulsion — external thrust versus internal mass ejection.
• Rigid Body Dynamics — CM as the point where $\vec{F}=M\vec{a}$ holds for extended bodies.