Angular Velocity Converter
Inputs
| Radian per second | 10 rad/s |
|---|
Angular Velocity Converter
Convert angular velocity between radians per second (rad/s), revolutions per minute (rpm), degrees per second (deg/s), and revolutions per second (rev/s).
Angular Velocity Units
Angular velocity
Angular velocity describes how fast an object rotates or revolves — the angle it sweeps out per unit of time. A spinning motor shaft, a bicycle wheel, a hard-drive platter, and a planet on its axis all have an angular velocity. The SI unit is the radian per second (rad/s), but engineering and everyday use rely on several other units for the same quantity.
Units of angular velocity
Radian per second (rad/s) — the SI unit. A radian is the angle subtended when the arc length equals the radius, and one full turn is 2π radians. So one revolution per second equals 2π rad/s ≈ 6.2832 rad/s. Physics equations for rotational kinematics and dynamics almost always use rad/s.
Revolution per minute (rpm) — turns per minute, the dominant unit for engines, motors, pumps, and turbines. Because one turn is 2π radians and one minute is 60 seconds, 1 rpm = 2π/60 rad/s ≈ 0.10472 rad/s.
Degree per second (deg/s) — the angle in degrees swept each second, common in navigation, gimbals, and computer graphics. A full turn is 360°, so 360 deg/s = 1 rev/s, and 1 deg/s = π/180 rad/s ≈ 0.017453 rad/s.
Revolution per second (rev/s) — full turns per second. It links the others cleanly: 1 rev/s = 2π rad/s = 360 deg/s = 60 rpm.
Conversion factors
| From | rad/s | rpm | deg/s | rev/s |
|---|---|---|---|---|
| 1 rad/s | 1 | 9.5493 | 57.2958 | 0.15915 |
| 1 rpm | 0.10472 | 1 | 6 | 0.016667 |
| 1 deg/s | 0.017453 | 0.16667 | 1 | 0.0027778 |
| 1 rev/s | 6.2832 | 60 | 360 | 1 |
The key anchors are 1 rev/s = 2π rad/s = 360 deg/s = 60 rpm and the handy shortcut 1 rpm = 6 deg/s.
Worked example
A motor is rated at 1,500 rpm and you need its angular velocity in rad/s for a torque calculation, plus deg/s for a control model.
- rpm to rad/s: multiply by 2π/60 ≈ 0.10472. 1,500 rpm × 0.10472 ≈ 157.08 rad/s
- rpm to deg/s: multiply by 6. 1,500 rpm × 6 = 9,000 deg/s
- rpm to rev/s: divide by 60. 1,500 rpm ÷ 60 = 25 rev/s
So 1,500 rpm ≈ 157.08 rad/s = 9,000 deg/s = 25 rev/s — one rotational rate expressed four ways.
Signed values
Angular velocity has a direction as well as a magnitude. A positive value conventionally means counter-clockwise rotation (right-hand rule) and a negative value clockwise. This converter accepts negative inputs, so entering −157.08 rad/s returns −1,500 rpm, preserving the sign across every unit.
Usage
Enter any value in any field. All other units update automatically. For example, type 60 in the rpm field to instantly see 6.2832 rad/s, 360 deg/s, and 1 rev/s.
Frequently Asked Questions (FAQ)
How do you convert rpm to rad/s?
Multiply the rpm value by 2π and divide by 60, because one revolution is 2π radians and one minute is 60 seconds. The factor is 2π/60 ≈ 0.10472. For example, 3,000 rpm × 0.10472 ≈ 314.16 rad/s. A quick sanity check: 60 rpm equals exactly 2π rad/s ≈ 6.2832 rad/s.
How do you convert rad/s to rpm?
Multiply the rad/s value by 60 and divide by 2π, the inverse of the rpm-to-rad/s factor. That gives 60/(2π) ≈ 9.5493 rpm per rad/s. For example, 10 rad/s × 9.5493 ≈ 95.49 rpm. Equivalently, 2π rad/s ≈ 6.2832 rad/s equals 60 rpm.
How do you convert deg/s to rpm?
One revolution is 360 degrees and one minute is 60 seconds, so 1 rpm = 6 deg/s. To convert degrees per second to rpm, divide by 6. For example, 90 deg/s ÷ 6 = 15 rpm. To go the other way, multiply rpm by 6 to get deg/s.
What is angular velocity?
Angular velocity measures how fast something rotates — the angle it sweeps per unit of time. Its SI unit is the radian per second (rad/s), where one full turn is 2π radians. Common alternatives are revolutions per minute (rpm) used for motors and engines, degrees per second (deg/s) used in navigation and graphics, and revolutions per second (rev/s). All express the same rotational rate at different scales.
Disclaimer
Conversions use exact definitions: 1 rev = 2π rad = 360°, so 1 rev/s = 2π rad/s = 360 deg/s = 60 rpm. This tool converts angular velocity (rotational rate) only and does not compute linear speed, which also depends on radius.