Angular Velocity
Simple Explanation
Angular velocity (Ο) measures how fast an object is rotating β the rate at which its angular displacement changes over time, in radians per second.
Why Do We Need It?
Angular velocity is the single number that describes "how fast something spins," independent of how big it is β a small wheel and a huge wheel turning at the same Ο complete a revolution in the same time.
Formula
Angular Velocity
Ο = ΞΞΈ / Ξt
The rate at which angular displacement changes over time β how fast something is rotating.
- Ο
- β angular velocity, in radians per second (rad/s)
- ΞΞΈ
- β change in angular displacement, in radians (rad)
- Ξt
- β time interval, in seconds (s)
When to use it: Whenever the rate of rotation needs to be found from the angle swept and the time it took.
Worked Example
Find an average angular velocity
A fan blade turns through 10 rad in 4 s. Find its average angular velocity.
Why Does This Work?
This mirrors ordinary velocity exactly β just as velocity is displacement divided by time, angular velocity is angular displacement divided by time, capturing how quickly the angle is changing.
Real-Life Example
How fast the Earth rotates
The Earth completes one full rotation (2Ο radians) approximately every 24 hours.
Dividing that angular displacement by the time gives the Earth's angular velocity β about 7.27Γ10β»β΅ rad/s β the same for every point on Earth's surface, from the equator to near the poles.
Practice
A wheel turns through 12 rad in 3 s. Find its average angular velocity.
EasyCommon mistake
Treating angular velocity (rad/s) as the same thing as linear velocity (m/s) β they measure different kinds of speed and are only related through v=rΟ, not equal to each other.
Quick Review
- Ο = ΞΞΈ/Ξt, in radians per second.
- Describes "how fast something spins," independent of size.
- The rotational counterpart of ordinary (linear) velocity.