What is Radian per second (rad/s)?
Radian per second (rad/s) is the unit of angular velocity. Angular velocity represents how much rotation an object undergoes per unit time and is measured in radians per second.
What is a Radian?
A radian is a unit of angular measure. The circumference of a circle is 2π times its radius; therefore a full circle contains 2π radians. 1 radian is the angle subtended at the center by an arc whose length equals the radius.
Radian per Second
Radian per second indicates how many radians an object rotates through in one second. For example, if an object has an angular velocity of 2 rad/s, it rotates through 2 radians every second.
Example
Consider a wheel completing one full revolution in one second. One full revolution equals 2π rad, so the angular velocity is ω = 2π rad/s. In general: ω = 2π·f and ω = 2π·(RPM/60).
Angular Velocity and Rotational Motion
Radian per second is used to express the rotational speed of wheels, propellers, drums, and even planets. Angular velocity is a fundamental characteristic of rotational motion. Related topic: What is Angular velocity?
Angular Velocity – Radians per Second (rad/s)
| Degrees (°) | Radians (in π) | Radians (≈) |
|---|---|---|
| 0 | 0 | 0.0000 |
| 30 | π/6 | 0.5236 |
| 45 | π/4 | 0.7854 |
| 60 | π/3 | 1.0472 |
| 90 | π/2 | 1.5708 |
| 120 | 2π/3 | 2.0944 |
| 135 | 3π/4 | 2.3562 |
| 150 | 5π/6 | 2.6180 |
| 180 | π | 3.1416 |
| 270 | 3π/2 | 4.7124 |
| 360 | 2π | 6.2832 |
Out in the field, we constantly convert nameplate RPM to rad/s for control calculations. The go-to formula is ω = 2π·(RPM/60). For a 1500 RPM motor, that’s roughly 157 rad/s. Keeping this in your pocket makes torque and speed discussions much clearer.
Students often find degrees intuitive and radians “too technical.” In rotating systems and control math, equations are written natively in radians, so reading trends, plots, and PID settings in rad/s becomes the practical habit.
Don’t forget the linear speed link: v = ω·r. With a larger drum radius, the same angular speed yields a higher belt or rim speed. If the conveyor’s belt speed is the target, you either pick ω or the drum diameter accordingly.
Bottom line: whether you are talking frequency–voltage on the electrical side or torque–inertia on the mechanical side, rad/s is the common language. Nail the basics and VFD settings, synchronous/asynchronous speed calculations, and feedback control all fall into place.
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