Angular Velocity to Linear Speed Calculator

Angular velocity describes how fast an object rotates (radians per second or RPM). Linear velocity describes how fast a point on that rotating object moves in a straight line. The relationship between them depends on the radius.

The Formula:

v = ω × r

Where:

• v = linear velocity (m/s)
• ω = angular velocity (radians per second)
• r = radius from the center of rotation (meters)

Converting RPM to rad/s:

ω (rad/s) = RPM × (2π / 60) = RPM × 0.10472

Worked Example:

A car tyre of rolling radius 0.33 m, which is a 15-inch rim with a normal-profile tyre on it, turning at 800 RPM:

ω = 800 × 0.10472 = 83.8 rad/s

Linear velocity = 83.8 × 0.33 = 27.6 m/s = 99 km/h

Note that the radius is the rolling radius of the whole tyre, not the rim size printed on the wheel. A 15-inch rim is 0.19 m in radius on its own; the rubber takes it to about 0.33. Use the rim figure and your answer comes out 40% low.

Centripetal acceleration comes free with the same numbers:

a_c = v² / r = ω² × r

For that tyre: 27.6² ÷ 0.33 = 2,316 m/s², about 236 g. That is what the rubber at the tread is pulling against the carcass at motorway speed, and it is why tyres have a speed rating at all. Double the speed and it quadruples.

Practical Applications:

Centripetal Acceleration:

A point on a rotating object also experiences inward acceleration:

a = ω² × r = v² / r

At the rim of a 0.33 m wheel doing 800 RPM: a = 83.8² × 0.33 = 2,318 m/s² ≈ 236g

Practical Tips:

• When calculating tire speed, use the loaded radius (slightly smaller than unloaded radius)
• Belt and pulley systems use this formula to calculate output speed from input speed
• A larger radius always produces higher linear velocity for the same angular velocity