What does this calculator do?
Angular velocity expresses how fast something rotates, in radians per second. It is the natural unit for rotational dynamics, motors, gears and control systems.
ω = RPM × 2π / 60
One revolution is 2π radians and one minute is 60 seconds, so you multiply RPM by 2π/60.
Worked example
1500 RPM is about 157.08 rad/s.
Practical tips
- Don’t mix up rad/s and RPM.
- In drivetrains, apply the gear ratio first if you need the speed of another shaft.
- For rotational mechanical power, use P = τω.
Common mistake to avoid
Keep the motor shaft speed and the speed after a gearbox clearly separate.
Complete angular velocity formulas
A full revolution is 2π radians and a minute has 60 seconds. Every conversion between RPM (revolutions per minute), rad/s and hertz follows from that:
- RPM to rad/s: ω = n · 2π / 60 ≈ n × 0.10472
- rad/s to RPM: n = ω · 60 / (2π) ≈ ω × 9.5493
- Rotational frequency: f = n / 60 (revolutions per second, Hz) and ω = 2π · f
- Time for one revolution: T = 1 / f = 60 / n
- Tangential speed at radius r: v = ω · r (m/s if r is in metres)
- Mechanical power: P = T · ω (W if the torque T is in N·m)
The radian is dimensionless, which is why rad/s can be written simply as s⁻¹ in dynamics equations.
RPM to rad/s conversion table
| RPM | rad/s | rev/s (Hz) | Typical reference |
|---|---|---|---|
| 60 | 6.283 | 1 | One revolution per second |
| 100 | 10.47 | 1.667 | Gearbox output |
| 1000 | 104.72 | 16.67 | Calculation reference |
| 1500 | 157.08 | 25 | Synchronous speed of a 4-pole motor at 50 Hz |
| 1750 | 183.26 | 29.17 | 4-pole motor at 60 Hz under load (approx.) |
| 3000 | 314.16 | 50 | 2-pole synchronous speed at 50 Hz |
| 3600 | 376.99 | 60 | 2-pole synchronous speed at 60 Hz |
The synchronous speed of an induction motor is n = 120 · f / p, where f is the supply frequency and p the number of poles. The actual speed under load is slightly lower because of slip.
Worked examples
Belt speed. A motor turns at 1450 RPM and drives a pulley with a 0.1 m radius. First, ω = 1450 × 2π / 60 ≈ 151.84 rad/s. The linear belt speed is then v = ω · r = 151.84 × 0.1 ≈ 15.18 m/s.
Power from torque. If the shaft delivers 9.55 N·m at 1500 RPM, P = T · ω = 9.55 × 157.08 ≈ 1500 W, that is, 1.5 kW. You can check it the other way round with the torque and power calculator.
Common mistakes
- Multiplying RPM by 2π without dividing by 60: the result would be in rad/min, 60 times too large.
- Confusing rad/s with Hz: 1 Hz of rotation (1 rev/s) is 2π ≈ 6.283 rad/s, not 1 rad/s.
- Using synchronous speed as the actual speed: a 4-pole induction motor on 60 Hz does not run at 1800 RPM under load but a little below; check the nameplate.
- Using the diameter in v = ω · r: the formula uses the radius; with the diameter you get twice the speed.
- Forgetting the transmission ratio: if there are gears or pulleys between the motor and the load, convert the speed first with the gear ratio calculator.
Frequently asked questions
How many rad/s is 1 RPM?
1 RPM = 2π / 60 ≈ 0.10472 rad/s. Conversely, 1 rad/s ≈ 9.549 RPM.
Why do physics formulas use rad/s instead of RPM?
Because with radians the relationships need no extra factors: v = ω · r and P = T · ω only work directly when ω is in rad/s. With RPM, constants such as 9550 appear in the torque formula.
Are angular velocity and angular frequency the same thing?
They are calculated the same way (ω = 2πf) and share units, but angular velocity describes a physical rotation while angular frequency describes an oscillating signal, such as a 60 Hz sine wave (ω ≈ 377 rad/s).
To understand the radian, angular acceleration and more examples, read the angular velocity guide.
Educational tool. Check tolerances, datasheets, standards and real operating conditions before using the results in critical applications.