Home › Calculators › Angular Velocity Calculator
Angular Velocity Calculator
Find angular velocity ω from an angle turned in a given time, from RPM, frequency or period, or from linear speed and radius. Results come in rad/s, degrees per second, RPM and Hz, with tangential velocity and centripetal acceleration when you add a radius.
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Optional except for linear speed; gives tangential speed and centripetal acceleration
Enter your values to see the result.
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How to use the Angular Velocity Calculator
- Choose the input method: angle and time, rotation rate, or linear speed and radius.
- Enter the values and pick units.
- Add a radius (optional) to get tangential speed and centripetal acceleration.
- Press Calculate to see ω in all units with the steps.
- Use the table to compare speeds at other radii on the same rotating object.
Angular velocity formulas
| From | Formula | Example |
|---|---|---|
| Angle and time | ω = Δθ ÷ Δt | 90° (1.5708 rad) in 2 s = 0.785 rad/s |
| RPM | ω = 2π · RPM ÷ 60 | 1,500 RPM = 157.08 rad/s |
| Frequency | ω = 2π · f | 50 Hz = 314.16 rad/s |
| Period | ω = 2π ÷ T | T = 60 s → 0.1047 rad/s |
| Linear speed | ω = v ÷ r | 10 m/s at r = 0.5 m → 20 rad/s |
Once you know ω, the speed of any point at radius r is v = ω · r and its centripetal acceleration is a = ω² · r = v² ÷ r.
Quick conversion table
| 1 unit | rad/s | °/s | RPM |
|---|---|---|---|
| 1 rad/s | 1 | 57.2958 | 9.5493 |
| 1 °/s | 0.017453 | 1 | 0.16667 |
| 1 RPM | 0.10472 | 6 | 1 |
| 1 Hz (rev/s) | 6.28319 | 360 | 60 |
Radians are the natural unit for rotation because one radian is the angle where the arc length equals the radius, so v = ωr works without extra factors only when ω is in rad/s.
Frequently asked questions
What is angular velocity?
Angular velocity ω is how fast something rotates: the angle turned per unit of time, ω = Δθ ÷ Δt. Its SI unit is radians per second (rad/s). Its direction follows the right-hand rule, and counterclockwise is usually taken as positive.
How do I convert RPM to rad/s?
Multiply by 2π and divide by 60: ω = RPM × 2π ÷ 60 ≈ RPM × 0.10472. So 1,500 RPM = 157.08 rad/s. To go back, RPM = ω × 60 ÷ 2π ≈ ω × 9.5493.
How do I find angular velocity from linear speed?
Divide the tangential speed by the radius: ω = v ÷ r, with v in m/s and r in meters. A car wheel of 13 inch (0.33 m) radius at 60 mph (26.8 m/s) spins at about 81 rad/s, or roughly 775 RPM.
What is the difference between angular velocity and angular frequency?
For steady rotation they have the same value, ω = 2πf, where f is the number of revolutions per second (Hz). Angular frequency is also used for oscillations such as springs and AC current, where nothing is physically rotating.
What is tangential velocity?
It is the linear speed of a point on a rotating object, v = ω·r. Points farther from the axis move faster, which is why the rim of a spinning disc travels faster than a point near the center even though both have the same angular velocity.
What is the angular velocity of the Earth?
The Earth turns once relative to the stars every 23.93 hours (a sidereal day), giving ω ≈ 7.29 × 10⁻⁵ rad/s. At the equator that corresponds to a tangential speed of about 465 m/s (1,040 mph).
Are all the tools on FreeToolCabin tested for quality and accuracy?
Yes! Every tool on our site undergoes automated Q/A testing in real browser environments to ensure high precision, correct output, and zero errors. If you happen to spot a bug or have a suggestion, please contact us at [email protected] and our team will review it right away.
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