📌 Quick Answer
The relationship between linear velocity (v) and angular velocity (ω) is v = ωr, where r is the radius of the circular path.
In words: the linear (tangential) speed of a point on a rotating body equals its angular velocity multiplied by its distance from the axis of rotation.
🔹 Key Takeaways
- v = ωr — linear velocity = angular velocity × radius.
- ω is measured in rad/s, v in m/s and r in metres.
- For the same ω, points farther from the axis have a higher linear velocity.
- The relation comes from the arc-length equation s = rθ.
Linear Velocity vs Angular Velocity
Linear (tangential) velocity (v) is the rate at which a point moves along its circular path, measured in metres per second (m/s). Angular velocity (ω) is the rate at which the body sweeps out angle about the axis, measured in radians per second (rad/s).
The Relationship: v = ωr (Derivation)
For a point moving on a circle of radius r, the arc length is s = rθ. Differentiating with respect to time t:
ds/dt = r · dθ/dt ⇒ v = ωr
because ds/dt is the linear velocity v and dθ/dt is the angular velocity ω. This single equation links the two.
Units and Conversions
ω must be in radians per second for v = ωr to give v in m/s. To convert from rpm: ω (rad/s) = 2πN/60, where N is revolutions per minute.
Linear vs Angular Velocity — Comparison
| Quantity | Symbol | Unit | Depends on radius? |
|---|---|---|---|
| Linear velocity | v | m/s | Yes (v = ωr) |
| Angular velocity | ω | rad/s | No (same for whole rigid body) |
Worked Example
A wheel of radius 0.5 m rotates at ω = 4 rad/s. The linear velocity of a point on its rim is:
v = ωr = 4 × 0.5 = 2 m/s.
Frequently Asked Questions
What is the relationship between linear and angular velocity?
Linear velocity equals angular velocity times the radius: v = ωr.
What is the formula connecting linear and angular velocity?
v = ωr, where v is linear velocity, ω is angular velocity in rad/s and r is the radius.
What are the units of angular velocity?
Angular velocity is measured in radians per second (rad/s).
Does linear velocity depend on the radius?
Yes. For the same angular velocity, a larger radius gives a larger linear velocity, since v = ωr.