Foundations — Rolling KE = ½mv² + ½Iω²

This note is the "prerequisite unlock" for the parent topic. Every letter, every squiggle, every idea the parent leans on is defined here — starting from a picture, never from another symbol you haven't met yet.

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1. Mass — the letter $m$ (and $M$)

The picture:

Look at the chalk wheel above. It is speckled with dots — each dot is one piece $m_i$. The parent note adds up energy piece by piece, so we must be able to name a single piece before we can add them.

Why the topic needs it: kinetic energy is built one particle at a time and then summed. Without the idea of "one piece $m_i$" there is nothing to sum.

Recall What does the little

$i$ mean under $m_i$? A label — "piece number $i$". It lets us talk about one specific chunk out of the many.

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2. The summation sign $\sum$

The picture: it's the big "add machine". You feed it a rule (like $\frac{1}{2}{m}_i{v}_i^2$) and it spits out the total after visiting every dot in the wheel.

Why the topic needs it: the whole derivation is "$KE={\sum }_i(\text{energy of piece }i)$". The $\sum$ is the tool that turns thousands of tiny energies into one number.

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3. Speed $v$, velocity $\vec{v}$, and the little arrow

The picture:

A blue arrow drawn from a moving dot: its length is the speed, its way it points is the direction. Speed is just the length of the arrow, ignoring where it points.

Why the topic needs it: the key move ${\vec{v}}_i={\vec{v}}_{cm}+{\vec{u}}_i$ (true motion = slide + spin) is an arrow addition. It only makes sense with directions attached.

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4. The centre of mass and its velocity ${\vec{v}}_{cm}$

The picture:

The wheel's hub is marked with a yellow dot; the yellow arrow is ${\vec{v}}_{cm}$, pointing forward along the ground. Every piece's motion is measured relative to this special point.

Why the topic needs it: König's theorem — the engine of the whole derivation — is literally "split every velocity into cm-motion + motion-seen-from-cm", and the clean split relies on ${\sum }_i{m}_i{\vec{u}}_i=0$.

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5. Relative velocity ${\vec{u}}_i$ (motion seen from the cm)

The picture: on figure s03, the pink arrows curling around the hub are the ${\vec{u}}_i$'s — pure spin, no forward drift, because we already subtracted the forward drift ${\vec{v}}_{cm}$.

Why the topic needs it: this is the exact line the parent derivation opens with. Each symbol in it is now built.

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6. The dot product $\vec{a}\cdot \vec{b}$

Why the topic needs it: using the distributive rule, expanding $({\vec{v}}_{cm}+{\vec{u}}_i)\cdot ({\vec{v}}_{cm}+{\vec{u}}_i)$ produces exactly three terms — ${\vec{v}}_{cm}\cdot {\vec{v}}_{cm}$ (slide), $2{\vec{v}}_{cm}\cdot {\vec{u}}_i$ (cross), ${\vec{u}}_i\cdot {\vec{u}}_i$ (spin). No distributive rule, no expansion.

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7. Angular velocity $\omega$

The picture: on figure s03 the pink circular arrow labelled $\omega$ shows the spin rate. A piece at distance $r_i$ from the hub has spin-speed $u_i=|{\vec{u}}_i|=\omega r_i$ — farther out means faster, like the tip of a clock hand outrunning its middle. (Here $u_i$ is the length of the spin arrow ${\vec{u}}_i$ from Section 5.)

Why the topic needs it: the spin energy is written in terms of $\omega$, and — once we add the rolling assumption below — $\omega$ gets tied to the slide speed. See Angular Velocity and Angular Acceleration for the full build.

Recall Why does a piece farther from the hub move faster during spin?

Because $u_i=\omega r_i$: the same spin rate $\omega$ sweeps a bigger arc when $r_i$ is larger.

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8. Distance from axis $r_i$ and radius $R$

The picture: figure s03 shows one pink piece with a dashed line of length $r_i$ back to the hub, and the full radius $R$ drawn straight down to the contact point.

Why the topic needs it: $r_i$ appears inside the spin-energy sum (each piece contributes $\frac{1}{2}{m}_i{r}_i^2{\omega }^2$); $R$ is the special value of $r_i$ for the rim piece touching the ground, and it is what links spin to slide in the next section.

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9. Rolling without slipping — the link $v_{cm}=\omega R$

Why the topic needs it: this single equation lets the parent rewrite the two-part KE using one speed, which is what makes the incline problems solvable.

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10. Moment of inertia $I_{cm}$ and the shape number $\beta$

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Prerequisite map

Each foundation feeds forward: pieces + sum build $M$ and $I$; the cm gives $\sum m_i{\vec{u}}_i=0$; that plus the dot-product distributive rule build the König split; no-slip adds $v_{cm}=\omega R$ and shape adds $\beta$; together they build the topic formula, which height + gravity then power on the incline.

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Equipment checklist