Linear and Angular Motion
Across the last four lessons, every rotational quantity has been introduced as a direct parallel to something already familiar from linear motion. That parallel isn't just a teaching device — it's the single most powerful problem-solving tool in this entire topic, and this lesson makes the whole dictionary of correspondences explicit in one place.
The linear–angular dictionary
Every quantity used to describe linear motion has a rotational counterpart that plays exactly the same mathematical role, just applied to rotation about an axis instead of motion along a line. The table below collects every correspondence met so far, plus two — moment of inertia and torque — that will be defined properly in the next topic but are included here to complete the picture.
| Linear quantity | Rotational quantity | Note |
|---|---|---|
| Displacement, $s$ (m) | Angular displacement, $\theta$ (rad) | Angle turned through, related by $s = r\theta$ |
| Velocity, $v$ (m s$^{-1}$) | Angular velocity, $\omega$ (rad s$^{-1}$) | Related by $v = r\omega$ |
| Acceleration, $a$ (m s$^{-2}$) | Angular acceleration, $\alpha$ (rad s$^{-2}$) | Related by $a = r\alpha$ (tangential) |
| Mass, $m$ (kg) | Moment of inertia, $I$ (kg m$^2$) | Covered fully in the next topic |
| Force, $F$ (N) | Torque, $T$ (N m) | Covered fully in the next topic |
| Momentum, $mv$ (kg m s$^{-1}$) | Angular momentum, $I\omega$ (kg m$^2$ s$^{-1}$) | Covered fully in the next topic |
The first three rows are exactly what's been built up across this topic. The last three foreshadow the next: just as force causes linear acceleration in proportion to mass ($F=ma$), torque will turn out to cause angular acceleration in proportion to moment of inertia, and just as momentum is mass times velocity, angular momentum will turn out to be moment of inertia times angular velocity. Recognising these correspondences now makes the next topic feel far less like new material and far more like the same ideas wearing rotational notation.
Connecting a rotating point to the whole rotating body
The angular quantities $\theta$, $\omega$ and $\alpha$ describe the rotation of a rigid body as a whole — every point on the body shares the same values. But any individual point on that body, sitting at radius $r$ from the axis, also has its own linear displacement, speed and acceleration, found by scaling the angular quantities by $r$:
The first two of these were derived explicitly in earlier lessons in this topic. The third, $a = r\alpha$, gives the tangential acceleration of a point at radius $r$ — the component of its acceleration directed along its direction of travel (tangent to the circular path), arising from the body's changing angular speed. This is distinct from centripetal acceleration, which is directed toward the axis and arises even at constant angular speed — the next lesson in this topic deals with that separately.
Whenever a rotational dynamics question feels unfamiliar, translate it into the linear picture first: replace $\theta$, $\omega$, $\alpha$ with their linear counterparts $s$, $v$, $a$ using $s=r\theta$, $v=r\omega$, $a=r\alpha$, solve the resulting problem with tools already second-nature from linear kinematics, then convert back if the question asks for an angular answer. Most exam questions in this topic are really linear-motion questions dressed in rotational notation — this dictionary is the key that unlocks them.
Key takeaways
- Every linear quantity has a rotational counterpart: $s \leftrightarrow \theta$, $v \leftrightarrow \omega$, $a \leftrightarrow \alpha$, and (introduced fully next topic) $m \leftrightarrow I$, $F \leftrightarrow T$, momentum $\leftrightarrow$ angular momentum.
- A point at radius $r$ on a rotating body has linear displacement, speed and tangential acceleration given by $s=r\theta$, $v=r\omega$, $a=r\alpha$.
- Tangential acceleration $a=r\alpha$ arises from changing angular speed, and is distinct from centripetal acceleration, which points toward the axis and exists even at constant angular speed.
- Translating a rotational problem into its linear equivalent, solving it with familiar linear-motion tools, then converting back, is the single most useful strategy across this whole topic.
- This dictionary of correspondences sets up centripetal acceleration and centripetal force, covered in the remaining lessons of this topic.