Angular Velocity converter

Every unit on this page is defined against the radian per second (rad/s), the SI unit of angular velocity. A radian is the angle swept when you travel one radius-length along a circle's edge, so radian per second measures how many of those arcs a rotating object sweeps through each second. The other units are just the radian rescaled for a friendlier interval or a friendlier angle: degree per second uses the degree, 1 of 360 slices of a circle (1°/s = 0.017453292519943295 rad/s), and revolution per minute (rpm) counts full 360° turns over a minute instead of a single second (1 rpm = 0.10471975511965977 rad/s). The per-hour and per-day variants simply stretch the same angle over a longer clock interval, which is why their factors shrink so quickly — a radian per day is thousands of times slower than a radian per second.

To convert revolution per minute to radian per second, multiply by that same factor: rad/s = rpm × 0.1047197551. The number comes from 2π radians per revolution divided by 60 seconds per minute, so it is not an arbitrary rounding — it is the exact ratio between the two definitions. It's also worth knowing that revolution per second is numerically identical to frequency expressed in hertz, since one full turn per second is exactly one cycle per second; that link is why rotating-machinery vibration analysis so often moves between rpm, hertz and radians per second within the same report.

A 4-pole AC induction motor running at its synchronous speed of 1,800 rpm turns at 1,800 × 0.1047197551 ≈ 188.5 rad/s. Going the other direction, a radar dish panning at 50°/s is sweeping 50 × 0.017453292519943295 ≈ 0.873 rad/s, which also works out to exactly 8.33 rpm — a handy check, since 360° of travel every second is one full revolution per second, or 60 revolutions per minute, and 50/360 of that is 8.33.

Angular velocity shows up wherever something spins at a measured rate: a car tachometer reads engine speed in rpm, a wind turbine's rotor typically turns a slow 10–20 rpm even though the generator it drives (through a gearbox) spins thousands of times faster, and a hard drive platter commonly spins at 5,400 or 7,200 rpm, a figure manufacturers chose specifically because it balances data-read speed against motor wear and noise. Drone flight controllers and inertial measurement units, by contrast, report raw gyroscope output in radians per second because that is what the underlying physics equations expect, and camera-gimbal or satellite-dish datasheets often quote slew rate in degrees per second because that is the unit a human operator can visualize most easily.

A common mix-up is treating rpm as if it were already an SI rate: because tachometers and spec sheets are so often labeled in rpm alone, it is easy to plug that number straight into a formula built for rad/s and get an answer 9.55 times too large. Also watch the per-day and per-hour units here — they describe genuinely slow rotations (a radian per day is about 4,000 times slower than a radian per second) and are meant for things like planetary rotation or glacier flow, not machinery, so a result in that range for a motor almost always means the wrong unit was selected.

Gear trains add one more wrinkle worth knowing: angular velocity doesn't survive a gear mesh unchanged the way linear speed does along a belt. Two meshed gears spin at angular velocities inversely proportional to their tooth counts, so a small gear driven by a large one always spins faster in rad/s or rpm, even though the teeth at the mesh point move at the same linear speed. Converting a gear train's output speed therefore means first finding the correct gear ratio, then applying the rad/s or rpm conversion on this page — the unit conversion and the mechanical ratio are two separate steps that are easy to blend together by mistake.