Angular Velocity: MDCAT Physics notes
Angular Velocity notes for MDCAT: ω = Δθ/Δt, rpm to rad/s, ω = v/r, period T = 2π/ω, clock hands and Earth, direction by right-hand rule.
Definition
Angular velocity is the rate of change of angular displacement:
$$\omega_{av} = \frac{\Delta\theta}{\Delta t}, \qquad \omega_{ins} = \lim_{\Delta t\to0}\frac{\Delta\theta}{\Delta t}$$
SI unit: $\text{rad s}^{-1}$. Other units: rev/s, rev/min (rpm), degrees per second.
Direction
Angular velocity is a vector along the axis of rotation, given by the right-hand rule: curl the fingers of the right hand in the sense of rotation; the thumb gives $\vec\omega$.
- Anticlockwise rotation on a page: $\vec\omega$ points out of the page, perpendicular to the plane.
- Clockwise: into the page.
- $\vec\omega$ is perpendicular to both the radius and the linear velocity, so the angle between $\vec v$ and $\vec\omega$ is always $90^\circ$.
Conversions
Since $1\ \text{rev} = 2\pi\ \text{rad}$:
- $\omega = 2\pi f$, where $f$ is in rev/s.
- $1\ \text{rpm} = 2\pi/60 \approx 0.105\ \text{rad s}^{-1}$.
- 400 rpm $= 400 \times 2\pi/60 = 40\pi/3 \approx 41.9\ \text{rad s}^{-1}$.
- 2 rev/s $= 4\pi \approx 12.6\ \text{rad s}^{-1}$.
- 12 rev in 4 s $= 3$ rev/s $= 6\pi \approx 18.8\ \text{rad s}^{-1}$.
Period and angular velocity
The time for one revolution is
$$T = \frac{2\pi}{\omega}$$
At $3.0\ \text{rad s}^{-1}$, $T = 2\pi/3 \approx 2.1\ \text{s}$.
| Rotating object | Period | Angular speed |
|---|---|---|
| Minute hand | 60 min | $\pi/30\ \text{rad min}^{-1} = \pi/1800\ \text{rad s}^{-1}$ ($6^\circ$ per minute) |
| Hour hand | 12 h | $\pi/6\ \text{rad h}^{-1}$ |
| Earth's spin | 24 h | $\pi/12\ \text{rad h}^{-1}$ ($15^\circ$ per hour) |
The Earth's orbit around the Sun (a revolution, motion around another body) takes about 12 months; spinning about its own axis is rotation.
Relation with linear speed
$$v = r\omega \quad\Rightarrow\quad \omega = \frac{v}{r}$$
- Aircraft at $250\ \text{m s}^{-1}$ on a 100 m radius turn: $\omega = 2.5\ \text{rad s}^{-1}$.
- Particle at $20\ \text{m s}^{-1}$, radius 10 m: $\omega = 2\ \text{rad s}^{-1}$.
In circular motion the direction of velocity keeps changing, so linear velocity is variable even at constant speed.
Angular kinematics
The equations mirror linear motion: $\omega_f = \omega_i + \alpha t$, $\omega_f^2 - \omega_i^2 = 2\alpha\theta$. From rest to $6\ \text{rad s}^{-1}$ at $2\ \text{rad s}^{-2}$: $\theta = 36/4 = 9\ \text{rad}$.
If a spinning body shrinks with no external torque, angular momentum $I\omega$ stays constant. For the Earth (a sphere, $I \propto R^2$), halving $R$ makes $I$ one quarter, so $\omega$ becomes 4 times and the day becomes 6 hours.
Key formulas
- $\omega = \Delta\theta/\Delta t = 2\pi f = 2\pi/T$
- $v = r\omega$
- $1\ \text{rpm} = 0.105\ \text{rad s}^{-1}$
- $\omega_f^2 - \omega_i^2 = 2\alpha\theta$
Common MDCAT traps
- Minute hand: $\pi/30$ rad per minute, not per second.
- Convert rpm by multiplying by $2\pi/60$.
- $\vec\omega$ is along the axis, not tangential or towards the centre.
- Uniform circular motion: speed constant, velocity variable.
Quick revision
- Anticlockwise: $\vec\omega$ out of the page.
- $\vec v \perp \vec\omega$.
- $T = 2\pi/\omega$.
- Minute hand sweeps $6^\circ$ per minute.
- Earth rotates $2\pi$ rad ($\approx 6.3$ rad) per day.