Angular Velocity
Angular velocity is the rate of change of angular displacement of a rotating object with respect to time. It is a vector quantity whose direction is given by the right-hand rule along the axis of rotation. Angular velocity is the rotational analogue of linear velocity and is central to the analysis of rotating machinery, celestial bodies, and rigid body dynamics.
Key Formula
ω = Δθ / Δt = 2π / T = 2πf
LaTeX: \omega = \frac{\Delta\theta}{\Delta t} = \frac{2\pi}{T} = 2\pi f
| Symbol | Meaning | Unit |
|---|---|---|
| ω | Angular velocity | Radian per second (rad/s) |
| Δθ | Change in angular displacement | Radian (rad) |
| Δt | Time interval | Second (s) |
| T | Period (time for one full rotation) | Second (s) |
| f | Frequency | Hertz (Hz) |
Worked Example
Problem
A ceiling fan completes 300 revolutions per minute (RPM). Calculate its angular velocity in rad/s and the linear speed of a blade tip 0.6 m from the axis.
Solution
Step 1: Convert RPM to rad/s. ω = 2π × (RPM/60) = 2π × (300/60) = 2π × 5 = 10π rad/s ≈ 31.4 rad/s Step 2: Find linear speed of blade tip. v = ω × r = 31.4 × 0.6 = 18.85 m/s
Answer
ω ≈ 31.4 rad/s; blade tip speed ≈ 18.85 m/s
Angular Velocities of Common Rotating Objects
| Object | RPM | Angular Velocity (rad/s) | Notes |
|---|---|---|---|
| Earth (rotation) | 0.000694 | 0.0000727 | One rotation per 24 hours |
| Ceiling fan (low) | 60 | 6.28 | Slow setting |
| Car engine (idle) | 700 | 73.3 | Crankshaft at idle |
| Car engine (max) | 7,000 | 733 | High-performance engine |
| Hard disk drive | 7,200 | 754 | Standard HDD spin rate |
| Dental drill | 300,000 | 31,416 | High-speed air turbine |
Interactive Tools
PhET Torque Simulation
Observe angular velocity and its relationship to torque and moment of inertia.
Open ToolDesmos Graphing Calculator
Graph angular displacement over time to visualize angular velocity.
Open ToolKhan Academy — Angular Velocity
Video lesson on angular velocity and its relationship to linear velocity.
Open Tool
Wikimedia Commons, CC BY-SA
Related Terms
Angular Acceleration
Angular acceleration is the rate of change of angular velocity with respect to time. Like its linear counterpart, it is a vector quantity and represents how quickly a rotating object is speeding up or slowing down its rotation. Angular acceleration is produced by a net torque and is related to it by the rotational analogue of Newton's second law.
Torque
Torque is the rotational equivalent of force — it is the tendency of a force to cause rotation about a pivot or axis. Mathematically, it is the cross product of the position vector (from the axis to the point of force application) and the force vector. Torque is essential in engineering design of engines, gears, wrenches, and any rotating machinery.
Moment of Inertia
Moment of inertia is the rotational analogue of mass — it measures an object's resistance to changes in its rotational motion about a given axis. It depends on both the total mass of the object and how that mass is distributed relative to the rotation axis; mass farther from the axis contributes more. Moment of inertia is fundamental in the design of flywheels, spinning tops, gyroscopes, and all rotating mechanical systems.
From Greek "angulos" meaning "angle" and Latin "velocitas" meaning "swiftness". The symbol ω is the Greek letter omega, traditionally used for angular quantities. The concept was developed alongside Newtonian rotational dynamics in the 17th century.