RPM to Radians per Second Converter

Convert revolutions per minute (RPM) to radians per second (rad/s) and back, with exact formulas and step-by-step examples.

RPM ⇄ rad/s Calculator

RPM

You can enter positive or negative values. Negative values indicate clockwise vs counter‑clockwise direction.

Result
0.1047
rad/s
1 RPM × 2π / 60
= 0.1047197551 rad/s

Quick RPM to rad/s reference

RPM rad/s (approx.)
1 0.1047
60 6.2832 (≈ 2π)
600 62.8319
3000 314.1593 (≈ 100π)
6000 628.3185 (≈ 200π)

How to convert RPM to radians per second

Revolutions per minute (RPM) measures how many full turns something makes in one minute. Radians per second (rad/s) is the SI unit of angular velocity and is used in most physics and engineering formulas.

RPM to rad/s formula

Each revolution is \(2\pi\) radians, and one minute is 60 seconds, so:

\[ \omega_{\text{rad/s}} = \text{RPM} \times \frac{2\pi}{60} \]

Numerically: \[ 1\ \text{RPM} \approx 0.1047197551\ \text{rad/s} \]

Step-by-step example: 3000 RPM to rad/s

  1. Start with the RPM value: \(3000\ \text{RPM}\).
  2. Multiply by \(2\pi\): \(3000 \times 2\pi = 6000\pi\).
  3. Divide by 60 to convert minutes to seconds: \[ \omega = 3000 \times \frac{2\pi}{60} = 100\pi\ \text{rad/s} \]
  4. Evaluate \(100\pi\): \[ 100\pi \approx 314.1593\ \text{rad/s} \]

So 3000 RPM ≈ 314.16 rad/s.

How to convert radians per second to RPM

To go the other way, divide by \(2\pi\) to get revolutions per second, then multiply by 60 to get revolutions per minute.

rad/s to RPM formula

\[ \text{RPM} = \omega_{\text{rad/s}} \times \frac{60}{2\pi} \]

Numerically: \[ 1\ \text{rad/s} \approx 9.549296586\ \text{RPM} \]

Example: 50 rad/s to RPM

  1. Start with the angular speed: \(50\ \text{rad/s}\).
  2. Multiply by 60: \(50 \times 60 = 3000\).
  3. Divide by \(2\pi\): \[ \text{RPM} = 50 \times \frac{60}{2\pi} = \frac{3000}{2\pi} = \frac{1500}{\pi} \approx 477.46\ \text{RPM} \]

So 50 rad/s ≈ 477.46 RPM.

Why radians per second matter in engineering

While RPM is intuitive for everyday use (motors, car engines, fans), most formulas in physics and engineering use radians per second:

  • Rotational kinetic energy: \(E = \tfrac{1}{2} I \omega^2\)
  • Torque–power relation: \(P = \tau \omega\)
  • Centripetal acceleration: \(a = \omega^2 r\)

In all of these, \(\omega\) must be in rad/s, not RPM. That’s why converting correctly is important when you move from specs to calculations.

Common RPM to rad/s conversions

  • 1 RPM ≈ 0.10472 rad/s
  • 10 RPM ≈ 1.0472 rad/s
  • 100 RPM ≈ 10.47198 rad/s
  • 1000 RPM ≈ 104.7198 rad/s

FAQ

Is RPM a unit of angular velocity?

Yes. RPM (revolutions per minute) is a unit of angular velocity, but it is not an SI unit. The SI unit is radians per second (rad/s). They describe the same physical quantity using different scales.

Can RPM be negative?

In many engineering contexts, a negative sign is used to indicate direction (clockwise vs counter‑clockwise). The magnitude of the angular speed is what matters for most formulas; the sign indicates orientation.

What is the exact conversion factor between RPM and rad/s?

The exact factor is:

  • \(1\ \text{RPM} = \dfrac{2\pi}{60}\ \text{rad/s} = \dfrac{\pi}{30}\ \text{rad/s}\)
  • \(1\ \text{rad/s} = \dfrac{60}{2\pi}\ \text{RPM} = \dfrac{30}{\pi}\ \text{RPM}\)