About the Angular Velocity Calculator
This angular velocity calculator finds how fast something rotates and expresses it in every common unit: radians per second (the SI unit), revolutions per minute (RPM), degrees per second and revolutions per second (hertz). You can start from a known rotation rate, from an angle turned in a given time, or from the linear speed of a point at a known radius, such as a car tyre rolling along the road.
Once ω is known, the calculator uses the radius to give the tangential (rim) speed of a point on the rotating object, the period of one revolution and the centripetal acceleration toward the centre. That makes it useful for physics homework on circular motion, for machinists checking surface speed, for drivers and cyclists relating wheel RPM to road speed, and for anyone sizing motors, pulleys, fans or turntables.
The calculator assumes steady rotation about a fixed axis, so ω is constant. Angular velocity is a vector whose direction follows the right-hand rule; here we report its magnitude, often called angular speed.
With the default inputs, the angular velocity ω is 314.159265 rad/s. Change any value above to recalculate instantly.
How to use the angular velocity calculator
- 1Choose whether you know a rotation rate, an angle and time, or a linear speed.
- 2Enter the value and pick its unit.
- 3Enter the radius to get tangential speed and centripetal acceleration.
- 4Read ω in rad/s along with RPM, °/s, Hz and the period.
Formula and method
Angular velocity ω is the angle swept per unit time, measured in radians per second. One full revolution is 2π radians, so multiplying revolutions per second by 2π (or RPM by 2π/60) converts to rad/s, and multiplying rad/s by 180/π gives degrees per second.
Every point on a rigid rotating body shares the same ω, but its linear speed grows with distance from the axis: v = ωr. Solving the same relation for ω gives the rotation rate from a known speed and radius. The centripetal acceleration needed to keep that point on its circle is ω²r (equivalently v²/r), and the period of one revolution is 2π/ω.
- ω
- Angular velocity (rad/s)
- Δθ, Δt
- Angle turned (rad) and time taken (s)
- f
- Rotational frequency (rev/s = Hz)
- r
- Radius from the axis (m)
- v
- Tangential (linear) speed (m/s)
- a_c
- Centripetal acceleration (m/s²)
- T
- Period of one revolution (s)
Worked examples
Motor spinning at 3,000 RPM with a 10 cm arm
3,000 RPM is 50 revolutions per second, so ω = 2π × 50 = 314.16 rad/s. A point 0.1 m out moves at 31.4 m/s and needs a centripetal acceleration of about 9,870 m/s², over 1,000 g.
Earth rotating once per sidereal day
One turn of 360° in 86,164 s gives ω = 7.292 × 10⁻⁵ rad/s. At the equatorial radius of 6,378 km, the ground moves at about 465 m/s, or 1,674 km/h.
Car tyre at 100 km/h
100 km/h is 27.78 m/s. Dividing by a rolling radius of 0.33 m gives ω = 84.2 rad/s, which is about 804 RPM, or 13.4 wheel revolutions every second.
Frequently asked questions
How do I convert RPM to rad/s?+
Multiply RPM by 2π and divide by 60, which is the same as multiplying by about 0.10472. For example, 1,000 RPM × 0.10472 = 104.72 rad/s. To go back, multiply rad/s by 9.5493.
What is the difference between angular velocity and angular speed?+
Angular speed is just the rotation rate, a positive number. Angular velocity is a vector: it has the same magnitude but also points along the rotation axis, with its direction given by the right-hand rule.
How is angular velocity related to linear velocity?+
For a point at distance r from the axis, linear (tangential) speed is v = ωr with ω in rad/s. Points farther out move faster even though the whole object turns at the same rate, which is why a wheel rim moves faster than its hub.
What is the angular velocity of the Earth?+
Earth turns once every sidereal day of about 86,164 seconds, so ω ≈ 7.292 × 10⁻⁵ rad/s, or about 15.04° per hour. At the equator that rotation gives a ground speed of roughly 465 m/s.
Is frequency the same as angular velocity?+
They measure the same rotation in different units. Frequency f counts revolutions per second (Hz); angular velocity counts radians per second. They are related by ω = 2πf.