Rotational Kinematics
Introduced| English | Meaning |
|---|---|
| angular velocity/ˈæŋɡjʊlə vəˈlɒsɪti/ | The rate of change of angular displacement, with a direction associated with the axis of rotation. |
| angular acceleration/ˈæŋɡjʊlə əkˌseləˈreɪʃn/ | The rate of change of angular velocity with time. |
| angular displacement/ˈæŋɡjʊlə dɪˈspleɪsmənt/ | The change in an object's angular position about a reference axis. |
| radian/ˈreɪdɪən/ | The angle subtended at a circle's centre by an arc equal in length to its radius. |
A spinning wheel needs a new set of words
- A car drives in a straight line; a wheel just spins in place. Both are "moving", but differently.
- To describe spinning we measure angles, not distances travelled.
- Three new quantities appear — angle, angular velocity, angular acceleration.
- They mirror position, velocity and acceleration exactly, so everything you learned carries over.
The three angular quantities
- Angular displacement 角位移 $\theta$ — the angle turned, measured in radians 弧度.
- Angular velocity 角速度 $\omega$ — how fast the angle changes, $\omega = \dfrac{\Delta\theta}{\Delta t}$ (in $\tfrac{\text{rad}}{\text{s}}$).
- Angular acceleration 角加速度 $\alpha$ — how fast $\omega$ changes, $\alpha = \dfrac{\Delta\omega}{\Delta t}$.
- These are the spin versions of $x$, $v$ and $a$.
A wheel goes from rest to $\omega = 12\ \tfrac{\text{rad}}{\text{s}}$ in $4\ \text{s}$. What is its angular acceleration, in $\tfrac{\text{rad}}{\text{s}^2}$?
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$\alpha = \Delta\omega/\Delta t = 12/4 = 3\ \tfrac{\text{rad}}{\text{s}^2}$.
Angular velocity is the rate of change of the ____.
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$\omega = \Delta\theta/\Delta t$ — the rate of change of the angle.
Radians, the natural angle
- A radian is the angle for which the arc length equals the radius.
- A full circle is $2\pi$ radians ($\approx 6.28$), so $180^\circ = \pi$ radians.
- Radians make the linking formulas (next lesson) clean and simple.
- Always work in radians for rotational physics, not degrees.
Sweep an angle in radians
Change the angle and radius to feel how radians measure the turn and the arc length.
How many radians are there in one complete turn?
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One full circle is $2\pi$ radians ($\approx 6.28$).
A half turn ($180^\circ$) is how many radians? Use $\pi = 3.14$.
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$180^\circ = \pi$ radians $\approx 3.14\ \text{rad}$.
The same equations, new symbols
- With constant angular acceleration, the SUVAT equations reappear, symbol for symbol.
- $\omega = \omega_0 + \alpha t$ mirrors $v = u + at$.
- $\theta = \omega_0 t + \tfrac12 \alpha t^2$ mirrors $s = ut + \tfrac12 a t^2$.
- Learn the analogy once and every rotation problem becomes a translation exercise.
Match each rotational quantity to its linear twin.
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$\theta \leftrightarrow x$, $\omega \leftrightarrow v$, $\alpha \leftrightarrow a$ — the rotational quantities mirror the linear ones.
Use radians, not degrees, in rotational formulas. A relation like $v = r\omega$ only works when $\omega$ is in $\tfrac{\text{rad}}{\text{s}}$. Mixing in degrees is one of the most common rotational-motion mistakes.
Rotational formulas like $v = r\omega$ require $\omega$ to be in radians per second.
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These relations are derived using radians; using degrees gives wrong answers.
A wheel starts from rest and reaches $\omega = 12\ \tfrac{\text{rad}}{\text{s}}$ in $4\ \text{s}$ at constant angular acceleration.
- $\alpha = \dfrac{\Delta\omega}{\Delta t} = \dfrac{12 - 0}{4} = 3\ \tfrac{\text{rad}}{\text{s}^2}$.
This is exactly like finding a linear acceleration — only the symbols changed.
Rotation is described by angular displacement $\theta$, angular velocity $\omega$ and angular acceleration $\alpha$ — the spin twins of $x$, $v$, $a$. Measure angles in radians ($2\pi$ per turn). With constant $\alpha$, the SUVAT equations reappear with $\theta,\omega,\alpha$.
Write from an angular acceleration of 2
- A wheel speeds up from rest to 10 rad/s in 5 s.
- Find the angular acceleration.
A wheel starts from rest and reaches 10 rad/s in 5 s. What is its angular acceleration, in rad/s squared?
10 divided by 5 is 2.
What is the angular acceleration?
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The angular speed rises by 10 rad/s in 5 s. Other options — 3 rad/s squared (the lesson wheel); 10 rad/s squared (the final speed used as an acceleration).