Angular Velocity Converter (rad/s, °/s, RPM)

Understanding the angular velocity converter — concept and principle

Angular velocity tells you how fast something turns: a motor shaft, a car wheel, a hard-disk platter, a turbine or a spinning figure skater. Engineers and physics textbooks use radians per second (rad/s), motor nameplates and machine shops use revolutions per minute (RPM), and sensors such as gyroscopes often report degrees per second (°/s). This converter moves a value between rad/s, °/s, RPM and revolutions per second, and shows the same value in every unit at once.

How it is calculated

Every unit is first turned into one common base, then divided by the size of the target unit: result = value × (size of "from" unit) ÷ (size of "to" unit). The fixed relationships are:

• 1 revolution = 2π rad = 360°
• 1 RPM = 2π/60 rad/s ≈ 0.1047198 rad/s = 6 °/s
• 1 rev/s = 60 RPM = 2π rad/s ≈ 6.283185 rad/s
• 1 rad/s = 60/(2π) RPM ≈ 9.549297 RPM ≈ 57.29578 °/s

The principle

A radian is the angle whose arc length equals the radius, so one full turn (circumference 2πr) is exactly 2π radians. That is why rad/s is the natural unit in physics: the speed of a point on the rim is simply v = ω·r, and mechanical power is P = torque × ω, both only when ω is in rad/s. RPM counts whole turns per minute, so going from RPM to rad/s means multiplying by 2π (radians per turn) and dividing by 60 (seconds per minute). Every factor here is exact by definition; the only rounding comes from π itself.

Worked example

Default input: 1,000 RPM to rad/s.

1. Radians per minute: 1,000 × 2π = 6,283.185 rad/min.
2. Divide by 60 s: 6,283.185 ÷ 60 = 104.7197551 rad/s.
3. In degrees: 1,000 × 360 ÷ 60 = 6,000 °/s.
4. In revolutions per second: 1,000 ÷ 60 = 16.66666667 rev/s.

How to read the result

Bigger numbers mean faster rotation, but the size of the number depends heavily on the unit: 1 rad/s is almost 10 RPM. Some reference points: a car engine idles at roughly 700–900 RPM; a 7,200 RPM hard disk turns at about 754 rad/s. Use rad/s whenever you plug the value into v = ωr, centripetal acceleration ω²r or P = Tω.

Common mistakes

• Using RPM directly in P = T·ω and getting a power about 9.55 times too large.
• Forgetting the 60: RPM is per minute, rad/s is per second.
• Labelling rev/s as "Hz" and then confusing it with angular frequency ω = 2πf.
• Mixing up angular velocity (rad/s) with linear speed at the rim (m/s), which also depends on the radius.

Frequently asked questions

How do I convert RPM to rad/s quickly?

Multiply by 0.10472 (2π/60). For a rough mental estimate, divide RPM by about 10.

Is rad/s the same as Hz?

No. Hz counts cycles per second. One cycle is 2π radians, so ω (rad/s) = 2π × f (Hz).

How do I get the rim speed of a wheel from RPM?

Convert to rad/s, then multiply by the radius in metres: v = ω × r gives metres per second.

Why do motor formulas use 9,549?

Power in kW = torque (N·m) × RPM ÷ 9,549, because 60,000/(2π) ≈ 9,549 combines the RPM-to-rad/s factor with watts-to-kilowatts.

Sources: BIPM, The International System of Units (SI Brochure), 9th edition, 2019 · NIST Special Publication 811 (2008), Guide for the Use of the International System of Units, Appendix B