Velocity - Angular Converter

Angular velocity describes how fast an object rotates about an axis — the rotational equivalent of linear velocity. Formally, angular velocity (ω, omega) is defined as the rate of change of angular position with respect to time: ω = dθ/dt. It is a vector quantity with direction along the axis of rotation according to the right-hand rule. Angular velocity is one of the most frequently encountered quantities in mechanical engineering, electrical engineering, robotics, physics, and astronomy.

The SI unit of angular velocity is the radian per second (rad/s). One radian per second means the rotating object sweeps through one radian of angle (approximately 57.296°) every second. A full revolution equals 2π ≈ 6.2832 radians, so 1 revolution per second = 6.2832 rad/s, and 1 revolution per minute (RPM) = 2π/60 ≈ 0.10472 rad/s. Rad/s is the unit required by physics equations, control system algorithms, and power calculations (P = τ × ω).

Revolutions per minute (RPM) is the most widely used practical unit for motors, engines, pumps, fans, turbines, compressors, and all rotating machinery. Typical values: car engine idle at 600–800 RPM, redline at 6000–8000 RPM. Industrial electric motors: 750–3600 RPM. High-speed machining spindles: 10,000–60,000 RPM. Computer hard disk drives: 5400 or 7200 RPM. Centrifuges: 3000–25,000 RPM. Dental drills: up to 400,000 RPM. Converting between RPM and rad/s is a daily necessity for engineers working with motors and power transmission.

Degrees per second (deg/s or °/s) is the standard unit in gyroscope and inertial measurement unit (IMU) specifications. Consumer-grade MEMS gyroscopes have full-scale ranges of ±250 to ±2000 deg/s. Racing drone flight controllers use gyroscopes rated to ±32,000 deg/s to handle extreme aerobatic maneuvers. Navigation-grade fiber-optic gyroscopes in aircraft and submarines measure rotation rates as small as 0.001 deg/h (0.0000028 deg/s).

Degrees per hour (°/h) and degrees per day (°/d) are used in astronomy, geodesy, and precision navigation. Earth rotates at exactly 15 degrees per hour (360° ÷ 24 h = 0.004167 °/s = 7.272 × 10⁻⁵ rad/s). The Moon orbits Earth at approximately 13.2 degrees per day. Satellite attitude drift rates, continental drift measurements, and long-period tidal oscillations are all expressed in these very small angular velocity units.

Revolution per second (r/s or rev/s) is used for high-speed machinery including centrifuges, precision spindles, and dental equipment. A laboratory centrifuge operating at 15,000 RPM = 250 rev/s = 1,570.8 rad/s.

Radian per day, hour, and minute appear in scientific applications where the SI radian base unit is preferred but time scales involve longer periods — such as satellite orbit analysis, tidal locking studies, and very long baseline interferometry (VLBI) measurements in radio astronomy. Foucault pendulums, which visually demonstrate Earth's rotation, precess at a rate expressible in radians per day that depends on latitude.

Angular velocity and torque together determine mechanical power: P = τ × ω. This equation requires angular velocity in rad/s when torque is in N·m and power is in watts. Engineers regularly convert RPM readings from tachometers into rad/s for power, efficiency, and dynamic analysis calculations. Gearbox ratio calculations also involve angular velocity ratios between input and output shafts.

This angular velocity converter supports all 12 units: radian/second, radian/day, radian/hour, radian/minute, degree/day, degree/hour, degree/minute, degree/second, revolution/day, revolution/hour, revolution/minute (RPM), and revolution/second — with 12-digit precision, instantly, completely free.