Angular Velocity Calculator
This angular velocity calculator converts rotational speed in rpm into angular velocity in radians per second and finds the linear speed at a given radius. Enter the rpm and the radius and the angular velocity calculator returns rad/s, metres per second, and frequency in hertz. It doubles as an angular speed calculator for motors, wheels, and shafts.
Results
The Angular Velocity Calculator is part of our Rotation collection. Browse more rotation to size, convert, and check your work alongside this tool.
What the angular velocity calculator shows
Angular velocity is how fast something turns, measured in radians per second. The angular velocity calculator converts the familiar rpm figure into rad/s and then finds how fast a point on the rim moves in a straight line. That makes it an angular and linear speed calculator in one step.
RPM to angular velocity calculator formula
The rpm to angular velocity calculator uses the fact that one revolution is 2π radians:
N is the speed in rpm, ω the angular velocity in rad/s, r the radius in metres, and v the linear speed. The frequency in hertz is N divided by 60.
Angular velocity to linear velocity calculator example
A shaft turning at 1,500 rpm with a point 0.1 m from the centre. The angular velocity to linear velocity calculator gives:
So a point 10 cm out moves at about 57 km/h.
How to use the angular speed calculator
Enter the rotational speed in revolutions per minute and the radius in metres from the centre to the point you care about. The angular speed calculator returns the angular velocity in rad/s, the linear speed at that radius in m/s, and the rotation frequency in Hz. For a wheel, use half the diameter as the radius.
Reading the angular velocity calculator result
Every point on a rigid rotating body has the same angular velocity, but the linear speed grows with distance from the centre. That is why the rim of a grinding wheel or a fan blade tip moves much faster than the hub. Use the linear speed to check surface speed limits for wheels and cutting tools.
When to use the rotational speed to linear speed calculator
- Finding blade tip speed with the rotational speed to linear speed calculator
- Converting motor rpm to rad/s for physics problems
- Checking wheel or pulley rim speed
- Working back with a linear speed to angular speed calculator
Common angular velocity calculator mistakes
- Forgetting 2π. One revolution is 2π radians, not 1.
- Using diameter for radius. Halve the diameter first.
- Mixing rpm and rev/s. Divide rpm by 60 for revolutions per second.
Tips for accurate results
- Keep the radius in metres for m/s output.
- Check tip speeds against the maker's limit.
- Use rad/s in physics formulas, rpm on machine labels.
Linear speed to angular speed calculator
Working the other way, a linear speed to angular speed calculator divides the linear speed by the radius: ω = v / r. A car wheel of 0.3 m radius at 25 m/s (90 km/h) turns at 83 rad/s, or about 796 rpm. The faster a body spins, the greater the inward force needed to hold it together, which the centripetal force calculator works out from mass, speed, and radius.
Calculate the angular velocity of the Earth about its axis
To calculate the angular velocity of the earth about its axis, divide one full turn, 2π radians, by the sidereal day of 86,164 seconds. The answer is about 7.29 × 10⁻⁵ rad/s. At the equator, with a radius of 6,378 km, that gives a surface speed of about 465 m/s.
Angular speed calculator reference
| Speed | Angular velocity | Frequency |
|---|---|---|
| 60 rpm | 6.28 rad/s | 1 Hz |
| 1,000 rpm | 104.7 rad/s | 16.7 Hz |
| 1,500 rpm | 157.1 rad/s | 25 Hz |
| 3,000 rpm | 314.2 rad/s | 50 Hz |
Angular Velocity Calculator FAQ
How do I convert rpm to rad/s?
Multiply the rpm by 2π and divide by 60. 1,500 rpm is about 157 rad/s.
What is the angular velocity of the Earth about its axis?
About 7.29 × 10⁻⁵ rad/s: one turn of 2π radians in a sidereal day of 86,164 seconds.
How do I get linear speed from angular speed?
Multiply the angular velocity in rad/s by the radius in metres to get metres per second.
Is angular velocity the same as rpm?
They measure the same thing in different units. rpm counts revolutions per minute; angular velocity uses radians per second.