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Relation between linear and angular quantities
Beginner
Tangential speed is ω times radius.
v = ωrFor a point at radius r: v = ωr, a_t = αr, a_c = ω²r. Points farther from the axis move faster.
A fan blade tip at 0.5 m with ω = 40 rad/s has v = 20 m/s.
These hold at each instant for a rigid body about a fixed axis.
In more detail
Linear and angular quantities are linked: s = rθ, v = rω and a_t = rα, where r is the distance from the axis. Each point on a rotating body follows these relations.
Real-world uses
- The speed of a point on a wheel's rim is the wheel radius times its angular velocity.
- Tyre speed ratings relate rotation and road speed.
- A cyclist knows the wheel's angular speed and radius, and uses v = ωr to find how fast the bike is moving.
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Frequently asked questions
Do all points on a rotating body have the same angular velocity?
Yes, for a rigid body. But their linear speeds differ depending on distance from the axis.
Which point is slowest?
The point on the axis, which has zero linear speed.
Does a point at the centre of a wheel have linear speed?
No. The centre is the axis, so its tangential speed is zero, while the rim has the highest speed.
Related topics in Rotational Motion
Angular velocity · Angular acceleration · Radians and degrees · Rotational kinematics · Torque · Lever arm and moment