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Angular displacement and velocity
Beginner
Angle swept and its rate of change.
ω = dθ/dtAngular displacement θ is in radians (arc length ÷ radius). Angular velocity ω = dθ/dt is in rad/s and is related to the linear speed by v = ωr.
A wheel turning at 120 rpm has ω = 120·2π/60 ≈ 12.6 rad/s.
Use radians in all formulas like v = ωr.
In more detail
Angular displacement is the angle swept by a line from the centre, and angular velocity is its rate of change. Angular velocity relates to linear speed by v = ωr.
Real-world uses
- Motor speed is quoted in revolutions per minute, which is an angular velocity.
- Wheel speeds in cars are linked to linear speed through the wheel radius.
- A cyclist's wheel spins at a known angular velocity, which tells the rider how many revolutions the wheel makes per second.
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Frequently asked questions
What units are used for angular velocity?
Radians per second, though revolutions per minute and hertz are also common.
How do angular and linear speed relate?
Linear speed equals angular velocity times the radius.
Why use radians?
Radians make formulas such as v = ωr simple, because the angle is the arc length divided by the radius, with no extra constants.
Related topics in Kinematics
River-boat problems · Rain and umbrella problem · Uniform circular motion · Period and frequency · Centripetal acceleration · Non-uniform circular motion