Angular Velocity Converter
Convert between rad/s, deg/s, rpm, Hz, and other angular velocity units.
Angular Velocity Converter
Enter an angular velocity and select its unit β all others update instantly with a live rotating diagram.
60 RPM
= 6.2832 rad/s
= 360 Β°/s
Quick References
How to Convert Between RPM, Rad/s, and Hz
A motor spec sheet says 3,600 RPM. Your physics formula needs rad/s. Type 3600, select RPM, and see that it equals 376.99 rad/s, 60 Hz, and 21,600 degrees per second. The live rotating diagram speeds up or slows down to match your input, giving you visual confirmation that the number makes sense.
The converter handles 8 units: rad/s, degrees/s, RPM, rev/s, Hz, degrees/min, degrees/hr, and arcsec/s. An astronomy student calculating Earth's rotation rate enters 15 degrees/hr and sees it equals 0.004167 deg/s, 7.27e-5 rad/s, or 0.000417 RPM. A vibration engineer measuring 120 Hz on a motor sees it equals 7,200 RPM and 753.98 rad/s. The quick-reference row shows clock hands, vinyl records, washing machines, and jet turbines for context.
Why the wrong unit gives wrong answers in formulas
The formula L = IΟ (angular momentum = moment of inertia x angular velocity) only works when Ο is in rad/s. A flywheel with I = 2 kg*mΒ² spinning at 600 RPM does NOT have L = 2 x 600 = 1200. You must convert first: 600 RPM = 62.83 rad/s. Then L = 2 x 62.83 = 125.66 kg*mΒ²/s. Using RPM directly gives an answer that is 9.55x too large.
A motor vibrates at 120 Hz. To convert to RPM: multiply by 60 = 7,200 RPM. To convert to rad/s: multiply by 2Ο = 753.98 rad/s. If you use 120 in a formula that expects rad/s, you are off by a factor of 6.28. If you use 7,200 in a formula that expects Hz, you are off by a factor of 60.
Earth rotates at 15 degrees per hour. In arcsec/s: 15 x 3600 / 3600 = 15 arcsec/s. A telescope mount must track at exactly this rate. If you program the mount motor using degrees/hr (15) instead of arcsec/s (15), the units are numerically equal but the conversion factor is wrong if you mix them. The converter eliminates this ambiguity.
How the Angular Velocity Converter Handles RPM, Rad/s, and Hz
All conversions pass through radians per second (rad/s). When you enter 60 RPM, the tool multiplies by (2Ο)/60 = Ο/30 = 0.10472 to get 6.2832 rad/s. Then it divides by each target unit's factor: by 1 for rad/s, by Ο/180 for degrees/s, by 2Ο for Hz, by 1/60 for rev/s. The result: 60 RPM = 6.2832 rad/s = 360 deg/s = 1 Hz = 1 rev/s.
Conversion factors derived from one revolution = 2Ο radians
The fundamental relationship is: 1 revolution = 2Ο radians = 360 degrees = 1 Hz (if per second). From this: rad/s to RPM multiply by 60/(2Ο) = 9.5493. RPM to Hz divide by 60. Degrees/s to rad/s multiply by Ο/180 = 0.01745. Arcsec/s to rad/s multiply by Ο/648,000 = 4.8481e-6. All factors are irrational (involving Ο), so they are stored as IEEE 754 double-precision approximations with 15-17 significant digits.
The live SVG animation and its timing
The rotating arm uses requestAnimationFrame to update at 60 fps. Each frame calculates the time delta dt in seconds, then adds Ο Γ dt to the current angle. The arm tip coordinates are computed using standard parametric circle equations: x = cx + r Γ sin(ΞΈ), y = cy - r Γ cos(ΞΈ). The animation speed is capped at 200 rad/s to prevent the arm from spinning too fast to see. The arc path shows the swept angle as a blue curve.
The converter handles the full range from arcsec/s (for astronomical tracking) to 50,000+ RPM (for centrifuges). Very small values use scientific notation (e.g., 7.27e-5 rad/s for Earth's rotation). Very large values use locale-formatted numbers. The dynamic formatting ensures readability whether you are working with clock hands (0.0001454 rad/s) or jet turbines (4,188.79 rad/s).
Frequently Asked Questions
How do I convert 3000 RPM to rad/s?
Multiply by Ο/30: 3000 x Ο/30 = 314.16 rad/s. Or divide by 9.5493: 3000 / 9.5493 = 314.16 rad/s. In Hz: 3000 / 60 = 50 Hz. This is a typical car engine speed. At 50 Hz, the engine completes 50 full revolutions per second, with each cylinder firing every other revolution.
What is the angular velocity of a clock's second hand?
The second hand completes one revolution (2Ο radians) in 60 seconds. So its angular velocity is 2Ο/60 = 0.1047 rad/s = 1 RPM = 0.0167 Hz. The minute hand is 60x slower: 0.001745 rad/s = 1/60 RPM. The hour hand is 12x slower than the minute hand: 0.0001454 rad/s = 1/720 RPM.
Why must physics formulas use rad/s instead of RPM?
Radian is a dimensionless ratio (arc length / radius). This makes rad/s compatible with formulas like torque (Ο = IΞ±), angular momentum (L = IΟ), and power (P = ΟΟ). If you use RPM directly, you get wrong answers. For example: P = Ο Γ Ο. If Ο = 10 N*m and Ο = 100 RPM, you cannot just multiply. Convert first: 100 RPM = 10.47 rad/s. Then P = 10 x 10.47 = 104.7 W.
What is the relationship between Hz and RPM?
1 Hz = 60 RPM. So divide RPM by 60 to get Hz. A 60 Hz motor spins at 3600 RPM. A 50 Hz motor spins at 3000 RPM. A washing machine at 1200 RPM = 20 Hz. A centrifuge at 50,000 RPM = 833.3 Hz. In the US, mains electricity is 60 Hz = 3600 RPM. In Europe, it is 50 Hz = 3000 RPM.
How do I calculate linear velocity from angular velocity?
Use v = rΟ, where Ο is in rad/s. A point on the rim of a 0.5 m radius disk spinning at 600 RPM has: Ο = 62.83 rad/s, v = 0.5 x 62.83 = 31.42 m/s = 113 km/h. For a car wheel with 0.35 m radius at 60 km/h: v = 16.67 m/s, so Ο = v/r = 16.67 / 0.35 = 47.62 rad/s = 454.8 RPM.
What is Earth's rotation rate in different units?
Earth rotates 360 degrees in 24 hours = 15 deg/hr = 0.004167 deg/s = 7.27e-5 rad/s = 0.000417 RPM = 1.157e-5 Hz. In arcsec/s: 15 arcsec/s (since 15 deg/hr = 15 x 3600 arcsec / 3600 sec = 15 arcsec/s). Telescope mounts must track at exactly this rate to compensate for Earth's rotation.