Angular Velocity Calculator
Angular velocity (ω) measures how quickly an object rotates or revolves around a point. It is expressed in radians per second (rad/s) and can be computed from a linear speed and radius (ω = v / r), from a period (ω = 2π / T), or from a frequency (ω = 2πf). This calculator accepts either a linear speed with radius, or a period or frequency, and returns the angular velocity in rad/s. It also shows the equivalent in revolutions per minute (RPM) and degrees per second, which are commonly used in engineering contexts.
Angular velocity formulas
ω = v / r | ω = 2π / T | ω = 2πf
Where v is linear speed (m/s), r is radius (m), T is period (s), and f is frequency (Hz). To convert to RPM: RPM = ω × 60 / (2π).
Understanding angular velocity
- A full circle is 2π radians (about 6.283), so one revolution per second equals 2π rad/s = 60 RPM.
- All points on a rigid rotating body share the same angular velocity, but their linear speeds differ with radius.
- Angular velocity is a vector quantity: its direction is along the axis of rotation (right-hand rule).
- Earth completes one rotation in 86,400 s, giving ω = 2π / 86400 = 7.27 x 10^-5 rad/s.
- Gears and pulleys multiply or divide angular velocity by their radius ratios.
Angular velocity: frequently asked questions
What is angular velocity?
Angular velocity (omega) describes how fast an object rotates, measured in radians per second (rad/s). A full rotation is 2pi radians, so an object completing one revolution per second has an angular velocity of 2pi rad/s (approximately 6.28 rad/s).
How do I convert RPM to rad/s?
Multiply RPM by 2pi and divide by 60: omega = RPM x 2pi / 60. For example, 1,000 RPM = 1000 x 2pi / 60 = 104.72 rad/s.
What is the relationship between linear speed and angular velocity?
Linear (tangential) speed v at radius r is related to angular velocity by v = omega x r, or equivalently omega = v / r. A point on the rim of a wheel moves faster than a point near the hub, but both have the same angular velocity.
What is the difference between angular velocity and angular frequency?
In mechanics angular velocity (omega) and angular frequency (omega) use the same symbol and formula (2pi f or 2pi / T), but angular frequency is used for oscillating systems while angular velocity is used for rotating objects. They are numerically identical.
Can angular velocity be negative?
Yes. The sign indicates the direction of rotation. By convention, counterclockwise rotation is positive and clockwise rotation is negative. This matters when dealing with vectors, torque, and angular momentum.
Official sources
- NIST Reference on Constants, Units, and Uncertainty: NIST Physics.
- OpenStax University Physics Volume 1: Rotational Variables.
Reviewed by the CalculatorHub team, edited by James Graham, 15 June 2026. See our methodology.