Angular Velocity Calculator

Calculate angular velocity ω from a rotation period, frequency, or rotations-over-time, via ω = 2π/T = 2πf, in rad/s, deg/s, RPM, and rev/s — plus optional tangential velocity v = ωr — with an animated rotating disk and a comparison scale from the Moon to a millisecond pulsar.

Angular velocity calculator

Angular velocity is angle swept per unit time: ω = Δθ/Δt, or for steady rotation, ω = 2π/T = 2πf. Every point on a rigid rotating body shares the same ω — but a point's tangential speed, v = ωr, grows with its distance from the axis.

Rotation period (T)

Radius (r) — optional, for tangential velocity

Radians/second7.292 × 10⁻⁵ rad/s

Degrees/second0.00418 °/s

Revolutions/minute (RPM)6.963 × 10⁻⁴ RPM

Revolutions/second1.161 × 10⁻⁵ rev/s

At radius 6378 km: tangential speed v = ωr ≈ 465.1 m/s (1674.3 km/h)

One full lap of the small dot = one rotation, at this ω. At radius 6378 km, that dot is moving at 465.1 m/s. Animation speed is compressed logarithmically to stay watchable — it preserves relative speed, not real time.

Log scale of ω in rad/s — spanning roughly nine orders of magnitude from the Moon's slow tidally-locked spin to a millisecond pulsar.

Anything that spins — a planet, a hard drive platter, a neutron star — shares a single number describing how fast it turns, regardless of its size: angular velocity. Two objects can have wildly different physical sizes and still turn at exactly the same rate, and this calculator converts between every common way of expressing that rate.

The relation

\omega = \frac{\Delta\theta}{\Delta t}

For steady rotation, one full turn is 2π radians, which gives the everyday form:

\omega = \frac{2\pi}{T} = 2\pi f

T is the rotation period (time for one full turn), f the rotational frequency (turns per unit time). Earth turns once every 23h 56m 4s (a sidereal day — slightly shorter than the familiar 24-hour solar day, because Earth also has to catch up with its own orbital motion around the Sun to bring the Sun back to the same spot in the sky) — that’s ω ≈ 7.29 × 10⁻⁵ rad/s. A 7200 RPM hard drive spins at ω ≈ 754 rad/s — roughly ten million times faster.

Same ω, very different speeds

Every point on a rigid rotating body shares that one ω. What doesn’t stay the same is tangential (straight-line) speed:

v = \omega r

A point twice as far from the axis moves twice as fast, in an ordinary meters-per-second sense, even though it completes exactly the same number of rotations per second as every closer point. This is why Earth’s equator (radius ≈6,378 km) is actually moving at about 1,674 km/h due to rotation alone, while a point near either pole is essentially standing still despite sharing the exact same ω.

When rotation gets extreme

Neutron stars make this concrete in a startling way. The Crab Pulsar spins about 30 times per second — already fast enough that a point on its ~10 km surface moves at roughly 0.6% of the speed of light. The fastest confirmed pulsar, PSR J1748-2446ad, spins 716 times per second — and a point on its surface would be moving at roughly 15% of the speed of light, purely from rotation. These are real measured values, not exaggerations; a neutron star’s small size is exactly what makes such extreme spin rates survivable at all — for a star-sized object, the same angular velocity would tear it apart, but for a body only tens of kilometers across, the actual (tangential) speeds involved stay merely relativistic rather than impossible.

Reading the visuals

The animated disk shows several points at different radii turning together — same ω, same animation, but the outer point visibly traces a bigger circle every rotation. Give it a radius and it prints the actual tangential speed at each point. The animation’s speed is compressed logarithmically so both a slow-turning Moon and a fast-spinning pulsar preset are watchable — it preserves relative speed (faster inputs animate faster), not real time.
The comparison scale places the current ω on a log scale next to everyday and astrophysical extremes — the Moon’s slow tidal lock, a washing machine, a hard drive, and both pulsar examples above — a span of roughly nine orders of magnitude on one axis.