# What is the angular velocity?

Angular velocity calculator: ω = Δθ ÷ Δt from an angle and time, ω = v ÷ r from a speed and radius, or from rpm or the time of one turn, in rad/s, °/s, rpm and Hz.

- Page: https://www.acalculator.org/physics/angular-velocity-calculator
- JSON spec: https://www.acalculator.org/physics/angular-velocity-calculator.json
- Version: dfd1c1307583

## Default answer

Example with the default inputs (From Angle and time, Angle turned (Δθ) 1,350, Angle unit rad, Time (Δt) 30 s, Radius (r) 0.328083989501312 ft): The angular velocity is 45 rad/s, or 429.718 rpm.

## Inputs

| Key | Label | Description |
| --- | --- | --- |
| by | From | What you know about the turning. |
| th | Angle turned (Δθ) | How far it turns, in the unit chosen below. |
| thu | Angle unit | The unit of the angle turned. |
| t | Time (Δt) | How long it takes to turn that angle. |
| v | Tangential speed (v) | The speed of a point on the circle along its path. |
| n | Rotation rate (n) | How fast it turns, in the unit chosen below. |
| nu | Rate unit | The unit of the rotation rate. |
| T | Time for one turn (T) | The period: how long one full turn takes. |
| r | Radius (r) | The distance from the axis to the point. Optional, except with a speed: gives the speed and acceleration there. |

## Outputs

| Key | Label | Description |
| --- | --- | --- |
| result | Answer | The angular velocity to 6 significant figures, in rad/s and rpm. |
| omega | Angular velocity (rad/s) | ω in radians per second. |
| degS | Angular velocity (°/s) | ω in degrees per second. |
| rpm | Revolutions per minute (rpm) | Full turns per minute: ω × 60 ÷ 2π. |
| hz | Turns per second (Hz) | Full turns per second, the frequency: ω ÷ 2π. |
| period | Time for one turn (s) | T = 2π ÷ ω, in seconds. |
| speed | Tangential speed (m/s) | v = r × ω at the radius, in metres per second. |
| accel | Centripetal acceleration (m/s²) | a = r × ω² at the radius, toward the axis. |

## Method

ω = Δθ ÷ Δt; ω = v ÷ r; ω = 2π × rpm ÷ 60; ω = 2π ÷ T; v = rω; a = rω².

## Assumptions

- The angular velocity is steady over the time given (an average otherwise).
- The radius is measured from the axis of rotation to the point that moves.

## Worked examples

1. by = angle, th = 1,350, thu = rad, t = 30, r = 0.1 gives omega = 45, v = 4.5, accel = 202.5, rpm = 429.718346, result = ω = 45 rad/s (429.718 rpm). Source: OpenStax, University Physics Volume 1, §10.1 Rotational Variables (Equations 10.3 and 10.4; Example 10.1: a flywheel at 45.0 rad/s turns 1,350 rad in 30 s, and a point 0.1 m out moves at 4.5 m/s; Example 10.2: 250 rpm is 26.2 rad/s), https://openstax.org/books/university-physics-volume-1/pages/10-1-rotational-variables (retrieved 2026-10-19); OpenStax, University Physics Volume 1, §6.3 Centripetal Force (a_c = v² ÷ r = rω²), https://openstax.org/books/university-physics-volume-1/pages/6-3-centripetal-force (retrieved 2026-10-19).
2. by = rate, n = 250, nu = rpm gives omega = 26.179939, hz = 4.166667, T = 0.24. Source: OpenStax, University Physics Volume 1, §10.1 Rotational Variables (Equations 10.3 and 10.4; Example 10.1: a flywheel at 45.0 rad/s turns 1,350 rad in 30 s, and a point 0.1 m out moves at 4.5 m/s; Example 10.2: 250 rpm is 26.2 rad/s), https://openstax.org/books/university-physics-volume-1/pages/10-1-rotational-variables (retrieved 2026-10-19).
3. by = speed, v = 30, r = 0.3 gives omega = 100, rpm = 954.929659, accel = 3,000. Source: OpenStax, University Physics Volume 1, §10.1 Rotational Variables (Equations 10.3 and 10.4; Example 10.1: a flywheel at 45.0 rad/s turns 1,350 rad in 30 s, and a point 0.1 m out moves at 4.5 m/s; Example 10.2: 250 rpm is 26.2 rad/s), https://openstax.org/books/university-physics-volume-1/pages/10-1-rotational-variables (retrieved 2026-10-19), Equation 10.4 (v_t = rω).
4. by = period, T = 86,164.09054 gives omega = 0.000073, degS = 0.004178. Source: NASA JPL Solar System Dynamics, Astrodynamic Parameters (mean sidereal day 86,164.09054 s), https://ssd.jpl.nasa.gov/astro_par.html (retrieved 2026-10-19); OpenStax, University Physics Volume 1, §10.1 Rotational Variables (Equations 10.3 and 10.4; Example 10.1: a flywheel at 45.0 rad/s turns 1,350 rad in 30 s, and a point 0.1 m out moves at 4.5 m/s; Example 10.2: 250 rpm is 26.2 rad/s), https://openstax.org/books/university-physics-volume-1/pages/10-1-rotational-variables (retrieved 2026-10-19).
5. by = angle, th = 90, thu = deg, t = 2 gives omega = 0.785398, degS = 45, T = 8. Source: OpenStax, University Physics Volume 1, §10.1 Rotational Variables (Equations 10.3 and 10.4; Example 10.1: a flywheel at 45.0 rad/s turns 1,350 rad in 30 s, and a point 0.1 m out moves at 4.5 m/s; Example 10.2: 250 rpm is 26.2 rad/s), https://openstax.org/books/university-physics-volume-1/pages/10-1-rotational-variables (retrieved 2026-10-19).

## FAQ

### How do you calculate angular velocity?

Divide the angle turned by the time: ω = Δθ ÷ Δt, in radians per second. A flywheel that turns 1,350 rad in 30 s has ω = 1,350 ÷ 30 = 45 rad/s.

### How do you convert rpm to rad/s?

Multiply by 2π (radians in one turn) and divide by 60 (seconds in a minute): ω = rpm × 2π ÷ 60. So 250 rpm = 26.18 rad/s, and 1 rad/s = 9.5493 rpm.

### What is the relation between angular velocity and linear speed?

A point at radius r moves along its circle at v = rω, so ω = v ÷ r. A wheel of radius 0.3 m rolling at 30 m/s turns at ω = 30 ÷ 0.3 = 100 rad/s, or 954.93 rpm.

### How do you find angular velocity from the period?

One turn is 2π radians, so ω = 2π ÷ T. The Earth turns once in a sidereal day of 86,164.09 s, so ω = 2π ÷ 86,164.09 = 7.2921 × 10⁻⁵ rad/s.

### What is the difference between angular velocity and angular frequency?

For steady rotation they have the same value, ω = 2πf, in rad/s. The frequency f is the number of turns per second, in hertz. Angular velocity also has a direction along the axis (by the right-hand rule).

### What is the centripetal acceleration at a radius?

a = rω², toward the axis. A point 0.1 m from the axis of a flywheel at 45 rad/s has a = 0.1 × 45² = 202.5 m/s², about 21 g.

## Sources

- OpenStax, University Physics Volume 1, §10.1 Rotational Variables: Equation 10.3 (ω = Δθ ÷ Δt), Equation 10.4 (v_t = rω); Example 10.1 (a flywheel at 45.0 rad/s turns 1,350 rad in 30 s; 4.5 m/s at 0.1 m); Example 10.2 (250 rpm is 26.2 rad/s), retrieved 2026-10-19. https://openstax.org/books/university-physics-volume-1/pages/10-1-rotational-variables
- OpenStax, University Physics Volume 1, §6.3 Centripetal Force: a_c = v² ÷ r = rω², retrieved 2026-10-19. https://openstax.org/books/university-physics-volume-1/pages/6-3-centripetal-force
- NASA JPL Solar System Dynamics, Astrodynamic Parameters: mean sidereal day 86,164.09054 s, retrieved 2026-10-19. https://ssd.jpl.nasa.gov/astro_par.html
- NIST Special Publication 811 (2008), Appendix B.8: 1 in = 0.0254 m, 1 ft = 0.3048 m, 1 mi = 1,609.344 m, 1 mph = 0.44704 m/s, 1 km/h = 1 ÷ 3.6 m/s, retrieved 2026-10-19. https://www.nist.gov/pml/special-publication-811