Foundations — Angular displacement θ, angular velocity ω, angular acceleration α

This page assumes you know nothing. We build every letter, ratio and picture the parent note Angular displacement, velocity, acceleration leans on, in an order where each idea rests on the one before it.

0. What is a "rigid body" spinning about an axis?

Before any symbol, fix the mental picture.

A rigid body is an object whose shape never changes — the distance between any two dots painted on it stays fixed forever. A spinning bicycle wheel, a merry-go-round, a clock hand: all rigid.

An axis is the fixed line the object turns around — the pin through the centre. Points on the axis don't move at all; points far from it move the most.

Why the topic needs this: because the whole trick of rotational mechanics is that measuring in metres is unfair — the red dot near the rim races while the dot near the hub crawls. We need a quantity they share. That quantity is the angle.

1. The circle, the radius $r$, and arc length $s$

Pick one point on the spinning body. As it turns, it traces a circle.

• $r$ = radius = straight-line distance from the axis to that point (in metres, m).
• $s$ = arc length = the curved distance the point actually travels along the circle (in metres, m).

Why we need both: $r$ is fixed for a given point; $s$ grows as the point keeps moving. The ratio between them is about to give us the angle.

2. The angle $\theta$ — and why radians

Here is the key move. Look at the wedge cut out by the moving point. How "open" is that wedge? That openness is the angle $\theta$ (Greek letter theta, just a name for "angle").

We could measure it in degrees, but there is a smarter unit built exactly for circles.

That definition is literally a ratio:

$\theta =\frac{s}{r}(\text{arc length}÷\text{radius})$

Why this ratio encodes the angle: a bigger wedge sweeps a longer arc $s$ for the same $r$, so the ratio $s/r$ grows. Dividing by $r$ removes the "how big is the circle" information, leaving only the openness — which is exactly what an angle is.

The whole circle: the arc all the way around is the circumference $s=2\pi r$. So a full turn is $\theta =\frac{2\pi r}{r}=2\pi \text{ rad}=360^{\circ }.$

3. What "rate of change" means (before any $\omega$ appears)

The parent note uses phrases like "rate of change" and symbols $\Delta$, $\frac{d}{dt}$. Let's earn them.

• $\Delta$ (Greek delta) means "change in" — a final value minus a starting value. $\Delta \theta ={\theta }_{\text{end}}-{\theta }_{\text{start}}$.
• $t$ = time, in seconds (s).
• A rate answers "how much does this change per second?" — you take the change and divide by the time it took: $\frac{\Delta \theta }{\Delta t}$.

The picture: plot $\theta$ up the page against time $t$ across it. Over an interval, the rate is the steepness of the line joining the two points — rise over run.

Why the topic needs it: angular velocity is just "rate of change of angle," and angular acceleration is "rate of change of that velocity." Both are steepnesses on a graph.

Why not just always use $\Delta$? Because the object's spin rate can vary from moment to moment. $\frac{d\theta }{dt}$ tells the truth right now; $\frac{\Delta \theta }{\Delta t}$ only gives an over-the-interval smear. When the spin rate is constant, the two are equal.

4. The Greek alphabet the topic uses

Three Greek letters carry the whole topic. They are just labels — here is how to say and read them.

• $\omega =\frac{d\theta }{dt}$ — how fast the angle grows. Units rad/s.
• $\alpha =\frac{d\omega }{dt}$ — how fast the spin rate itself grows. Units rad/s².

Why they exist: they are the angular twins of position $x$, speed $v$, and acceleration $a$ — the same three ideas, but measured in angle instead of metres, so that all points of the rigid body share one set of numbers.

5. The radius as the bridge back to metres

Every point shares $\theta ,\omega ,\alpha$. But the point near the rim still feels faster. Where does that come from? The radius.

Rearranging $\theta =s/r$ gives the master bridge:

$s=r\theta .$

Multiply an angular quantity by $r$ and you land back in the linear (metres) world:

• distance: $s=r\theta$
• speed: $v=r\omega$
• along-path acceleration: $a_t=r\alpha$

The parent note derives $v=r\omega$ and $a_t=r\alpha$ by differentiating $s=r\theta$ — that's just "take the rate of change of both sides," using the $\frac{d}{dt}$ tool we defined in §3.

6. One more distinction the topic assumes: tangential vs centripetal

A point on a circle can accelerate in two totally different directions:

• Tangential — along the path, speeding the point up or down. This is $a_t=r\alpha$.
• Centripetal — pointing inward to the centre, only bending the path (changing direction, not speed). This is $a_c={\omega }^{2}r$.

They are perpendicular. We flag this now only so the symbols $a_t$ and $a_c$ aren't a surprise later; the full story lives in Centripetal Acceleration and Force.

7. How these foundations feed the topic

Equipment checklist

Cover the right side and test yourself — you're ready for the parent note when each answer comes instantly.

Connections

• Parent topic — where these symbols are put to work.
• Linear Kinematics Equations — the $x,v,a$ world these angular quantities mirror.
• Uniform Circular Motion — the case $\alpha =0$, constant $\omega$.
• Centripetal Acceleration and Force — the inward $a_c={\omega }^{2}r$ we flagged in §6.