Foundations — Relation to linear quantities - v = rω, a_t = rα, a_c = rω²

Before you can trust those three formulas, you must own every letter inside them. This page builds each symbol from absolute zero — plain words first, then a picture, then why the topic can't live without it. Read top to bottom; each block leans on the one above.

1. The axis and a rigid body

The picture (Figure 1): imagine a flat disk pinned through its center by a nail. The nail is the axis. Pick any point on the disk — as the disk turns, that point traces a perfect circle centered on the nail. Two different points trace two different-sized circles, but they never collide or drift, because the body is rigid.

Why the topic needs it: "rigid" is the silent hero. It guarantees that each point's distance $r$ from the axis is constant in time. In every derivation on the parent page, we pull $r$ out of a derivative — that move is only legal because rigidity keeps $r$ fixed.

See also Angular velocity and angular acceleration where this same shared-rotation idea returns.

2. The radius $r$

The picture: in Figure 1, look at the straight spoke from the nail out to a point. Its length is $r$. A point at the rim has a big $r$; a point near the nail has a small $r$; a point ON the axis has $r=0$.

Why the topic needs it: $r$ is the whole dictionary. The core message "farther = faster" is literally $v=r\omega$ — bigger $r$, bigger $v$, for the same turning rate. Without a per-point $r$, angular and linear language could never be linked.

3. Angle $\theta$ and radian measure

We insist on measuring $\theta$ in radians, not degrees. Here is why, built from scratch.

The picture (Figure 2): take the circular path of a point. Lay the straight length $r$ down as a curved measuring tape along the arc. If exactly one $r$ fits, that's $1$ radian ($\approx 57.3°$). A full circle's arc is $2\pi r$ long, so a full turn is $2\pi$ radians $=360°$.

Why the topic needs radians: rearranged, the definition gives ==$s=r\theta$== — arc length equals radius times angle — but only in radians. In degrees you'd carry ugly factors of $\pi /180$ everywhere. Radians are engineered so this reads cleanly, and $s=r\theta$ is literally the seed the parent page differentiates to get every other formula.

Prerequisite deep-dive: Radian measure and arc length.

4. Arc length $s$ — the linear position along the path

The picture: in Figure 2, $s$ is the length of the curved trail the point leaves behind. It's a linear (metre) quantity, unlike $\theta$ which is an angular (radian) quantity.

Why the topic needs it: $s$ is the linking variable. It is angular on one side ($=r\theta$) and linear on the other (its rate of change is speed). That double life is exactly what lets $r$ translate between the two worlds.

5. Rate of change: reading a slope over time

Every derivation on the parent page uses the phrase "differentiate with respect to time." You must picture what that means before the symbols arrive.

The picture (Figure 3): plot any quantity against time. At an instant, draw the tangent line touching the curve. Its steepness (rise over run) is the rate of change at that moment. Steep upward line = growing fast; flat line = not changing; downward = shrinking.

Why the topic needs it — and why THIS tool: "speed" and "acceleration" are rates. To turn a position into a speed, or a speed into an acceleration, we need the one machine that reports instantaneous rate-of-change — that machine is the time-derivative. No other tool answers "how fast is this changing right now?" That is why the parent note differentiates $s=r\theta$ once (to get speed) and twice (to get acceleration).

6. The angular quantities: $\omega$ and $\alpha$

Now that "rate of change" has a picture, the angular quantities are easy.

The picture: on the merry-go-round, $\omega$ is the single "turning speed" everyone shares — one number for the whole disk. $\alpha$ is nonzero only when someone is cranking the disk faster (or braking it). If the spin is steady, $\alpha =0$.

Why the topic needs them: these are the inputs to the translator. Multiply $\omega$ by $r$ and you get linear speed $v$; multiply $\alpha$ by $r$ and you get tangential acceleration $a_t$. They are shared across the whole body, which is what makes angular language so simple.

Full treatment: Angular velocity and angular acceleration.

7. The linear quantities: $v$, $a_t$, $a_c$

These are what the formulas produce — the private, per-point outputs.

The picture (Figure 4): at one point on the circle, draw three arrows — $\vec{v}$ tangent (grazing the circle), ${\vec{a}}_t$ along that same tangent line, and ${\vec{a}}_c$ aimed at the center. The tangent arrows are perpendicular to the inward arrow.

Why the topic needs them: Newton's laws, force, and energy all speak linear. To use them on a spinning body we must convert the shared angular quantities into each point's linear ones — that is the entire purpose of $v=r\omega$, $a_t=r\alpha$, $a_c=r{\omega }^2$. See Centripetal force and circular motion and Uniform vs non-uniform circular motion.

8. Sines and cosines — the direction bookkeepers

The centripetal derivation writes the point's position as $\vec{r}=r(\cos \theta ,\sin \theta )$. You must know what $\cos$ and $\sin$ mean here.

The picture: as $\theta$ grows from $0$, the point walks counterclockwise; its shadow on the horizontal axis is $\cos \theta$ (starts at $1$, shrinks), its shadow on the vertical axis is $\sin \theta$ (starts at $0$, grows). Multiplying by $r$ scales the unit circle up to the real circle.

Why the topic needs them: they encode direction as numbers, so we can differentiate the position vector and watch its direction rotate. That direction-tracking is exactly what reveals centripetal acceleration — a fact invisible if you only track speed.

Prerequisite map

Everything funnels into the parent topic.

Quick self-tests

Predict, then reveal.

Equipment checklist

Test yourself — reveal only after answering aloud.