Angular Velocity Calculator

RPM tells you how many full turns happen in a minute, but physics likes radians per second. So we convert. Each full turn is 2π radians, and a minute is 60 seconds, which gives us ω = RPM × 2π / 60 in rad/s. If you want degrees per second instead, multiply RPM by 6 (since one revolution is 360°, divided by 60 seconds). Quick example: a fan running at 300 RPM gives ω = 300 × 2π / 60 ≈ 31.42 rad/s. Always plug in the number first, then let the unit chase you.

Just punch in the RPM value and the calculator does the converting for you. Internally it applies the relation 1 RPM = 2π / 60 rad/s, which works out to roughly 0.10472 rad/s. So 100 RPM becomes about 10.47 rad/s, and 1500 RPM (a typical washing machine spin) is around 157 rad/s. Use this when a problem mixes engine specs (given in RPM) with formulas like v = ωr or centripetal force, which need rad/s. Saves you from sign errors and clunky unit juggling during exams.

Two inputs go in: linear velocity v of the point on the circle, and the radius r from the centre. The tool returns ω using ω = v / r. Keep your units honest — v in metres per second, r in metres, and you'll get ω in rad/s. If a wheel of radius 0.3 m rolls so its rim moves at 6 m/s, then ω = 6 / 0.3 = 20 rad/s. A common trap is mixing centimetres with metres, so convert before you click compute.

Frequency f tells you how many full cycles happen each second, measured in hertz. One full cycle is 2π radians, so the angular velocity comes out to ω = 2πf, with units of rad/s. A 50 Hz mains supply, for instance, gives ω = 2π × 50 ≈ 314.16 rad/s. This formula sits behind almost every oscillation problem you'll meet, from simple harmonic motion to AC circuits. The only thing to watch is that f must be in hertz, not RPM or RPS — those need their own conversion first.

The period T is just the time for one full revolution or oscillation. Since one revolution covers 2π radians, the angular speed becomes ω = 2π / T, expressed in rad/s. So a pendulum with a 2 second period swings with an angular speed of 2π / 2 ≈ 3.14 rad/s. Watch the units: T must be in seconds. If a question gives you milliseconds or minutes, convert first or you'll be off by orders of magnitude. This formula is the inverse cousin of ω = 2πf, since f = 1/T.

Linear velocity v is how fast a point moves along its path, in m/s. Angular velocity ω is how fast the angle sweeps, in rad/s. They connect through v = ωr, where r is the distance from the rotation axis. Two children on a merry-go-round share the same ω, but the one further from the centre has a bigger v — that's why the outside feels faster. Rearrange as ω = v/r when you know the linear speed. This relation is the bridge between rotational and translational motion.

60 RPM is one of the cleanest numbers to work with because one revolution every second comes out to ω = 2π rad/s ≈ 6.2832 rad/s. The arithmetic is 60 × 2π / 60, where the sixties cancel and you're left with 2π. I use this example often in class because it builds intuition: at 60 RPM, the second hand of a clock and a record player at the right speed share the same angular velocity. Once you remember 60 RPM = 2π rad/s, scaling to any other RPM is straightforward.