Rotational Kinematics handled spinning with constant angular acceleration. Everything you did in Velocity and Position from Acceleration carries over to rotation with one change of letters: $x \to \theta$, $v \to \omega$, $a \to \alpha$.
Key idea
$$\omega = \dfrac{d\theta}{dt}, \qquad \alpha = \dfrac{d\omega}{dt} = \dfrac{d^2\theta}{dt^2}, \qquad \omega(t) = \omega_0 + \int_0^t \alpha\,dt'.$$ Angles in radians, $\omega$ in rad/s, $\alpha$ in rad/s². Every point of a rigid body shares the same $\theta$, $\omega$ and $\alpha$.
Why it works
A radian is defined as arc length divided by radius, $\theta = s/r$, so it is a pure number and $s = r\theta$ holds exactly. Differentiating gives $v = r\omega$ and $a_t = r\alpha$ for a point at distance $r$ from the axis. Those simple links, and the formulas for rotational energy and work built on them, only hold with angles measured in radians.
Worked example: Differentiate the angle
A turntable's angle is $\theta(t) = 2t^3 - 3t^2$ (rad, s). Find $\omega$ and $\alpha$ at $t = 2\ \text{s}$.
- $\omega = \dfrac{d\theta}{dt} = 6t^2 - 6t$, so $\omega(2) = 24 - 12 = 12\ \text{rad/s}$.
- $\alpha = \dfrac{d\omega}{dt} = 12t - 6$, so $\alpha(2) = 18\ \text{rad/s}^2$.
Answer: $\omega = 12\ \text{rad/s}$, $\alpha = 18\ \text{rad/s}^2$.
Worked example: Integrate the angular acceleration
A wheel spinning at $3\ \text{rad/s}$ has $\alpha(t) = 4 - 2t$ (rad/s², s). Find $\omega$ and the angle turned at $t = 3\ \text{s}$.
- $\omega(t) = 3 + \int_0^t (4 - 2t')\,dt' = 3 + 4t - t^2$, so $\omega(3) = 6\ \text{rad/s}$.
- $\theta(t) = \int_0^t \omega\,dt' = 3t + 2t^2 - \tfrac{1}{3}t^3$, so $\theta(3) = 9 + 18 - 9 = 18\ \text{rad}$, almost 3 turns.
Answer: $\omega = 6\ \text{rad/s}$; $\theta = 18\ \text{rad}$.
Common mistake
Differentiating an angle given in degrees and calling the result rad/s. If $\theta = 30t^2$ degrees, then at $t = 2\ \text{s}$, $d\theta/dt = 120$ degrees per second, which is $120 \times \pi/180 = 2.09\ \text{rad/s}$, not 120 rad/s. Check: before using $v = r\omega$, $K = \tfrac{1}{2}I\omega^2$ or $P = \tau\omega$, make sure the angle was in radians; multiply degrees by $\pi/180$ and revolutions by $2\pi$.
| Straight line | Rotation | Link for a point at radius $r$ |
|---|---|---|
| $x$ (m) | $\theta$ (rad) | $s = r\theta$ |
| $v = dx/dt$ | $\omega = d\theta/dt$ | $v = r\omega$ |
| $a = dv/dt$ | $\alpha = d\omega/dt$ | $a_t = r\alpha$, and also $a_c = r\omega^2$ toward the axis |
Angular velocity is really a vector, $\vec{\omega}$, pointing along the axis: curl the fingers of your right hand the way the object turns, and your thumb points along $\vec{\omega}$. In fixed-axis problems we keep just its sign, but the vector form is what makes Torque as a Cross Product and Gyroscopes and Precession work.
Revolutions per minute (rpm) are common on motors: $1\ \text{rpm} = \dfrac{2\pi}{60}\ \text{rad/s} \approx 0.105\ \text{rad/s}$.