Angular Velocity Calculator

Using the angular velocity calculator

Angular velocity tells you how fast something turns around a centre: how much angle it sweeps out per unit of time. The calculator offers four calculations, selected with the options at the top of the form.

• Angular velocity via time to turn angle: enter the angle turned and the time it took.
• Angular velocity via linear velocity: enter the speed along the path and the radius of the path.
• Linear velocity: enter the angular velocity and the radius to get the speed along the circle in six units.
• Path radius: enter the angular velocity and the linear velocity to get the radius in four units.

In the first two modes, the "Output in" selectors choose the angle unit (radians, degrees or revolutions) and the time unit (per second up to per century). Longer periods are handy for slow motion such as planets and tides.

Conversions involving years or centuries use a year of 365.2421 days (31,556,917.44 seconds). A week is exactly seven days.

Angular velocity equations

Angular velocity, written ω (omega), is the angle swept divided by the time taken. With the angle in radians the result is in radians per second:

ω = θ ÷ t

When a body travels around a circle of radius r at a linear speed v, the two quantities are tied together by the radius:

v = ω × r

ω = v ÷ r r = v ÷ ω

Angular velocity is also linked to the frequency f (turns per second) and the period T (seconds per turn):

ω = 2π × f = 2π ÷ T

One full turn is 2π radians, 360 degrees or one revolution, so a quick conversion table is useful:

In the other direction, 1 rad/s ≈ 57.2958 °/s ≈ 9.5493 rpm.

Angular speed calculation examples

From angle and time. A turntable arm sweeps 90° in 2 seconds. In radians that is 90 × π ÷ 180 = π ÷ 2 ≈ 1.5708 rad, so ω = 1.5708 ÷ 2 ≈ 0.7854 rad/s, which equals 45°/s.

From linear velocity. A point on a rotating platform moves at 20 m/s along a circle of radius 3 m. ω = 20 ÷ 3 ≈ 6.667 rad/s, which is 381.97°/s, 1.061 revolutions per second or 63.66 rpm.

A car wheel. At 100 km/h (27.78 m/s) a wheel of radius 0.3 m turns at ω = 27.78 ÷ 0.3 ≈ 92.59 rad/s. Dividing by 2π and multiplying by 60 gives about 884 rpm.

Linear velocity from rpm. A disc of radius 0.5 m spins at 12 rpm. Then ω = 12 × 2π ÷ 60 ≈ 1.2566 rad/s and v = 1.2566 × 0.5 ≈ 0.6283 m/s, or 2.262 km/h.

Path radius. A body turns at 2.5 rad/s while moving at 60 km/h (16.67 m/s), so r = 16.67 ÷ 2.5 ≈ 6.667 m.

The Earth. One turn relative to the stars takes a sidereal day, about 86,164 seconds, so ω ≈ 2π ÷ 86,164 ≈ 7.29 × 10^-5 rad/s. At the equator, where the radius is about 6,378 km, that corresponds to a speed of roughly 465 m/s.

For comparison, the minute hand of a clock turns at 2π ÷ 3,600 ≈ 0.001745 rad/s (0.1°/s) and the hour hand at about 0.000145 rad/s.

Assumptions and limits

• The calculator assumes steady circular motion, so the angular velocity does not change during the interval. For speeding up or slowing down, the answer is an average.
• Only the size of the angular velocity is reported. The direction (clockwise or anticlockwise, or the axis of rotation) is not part of the result.
• All inputs must be positive. Zero and negative values are not accepted.
• Every point of a rigid rotating object has the same angular velocity, but points farther from the axis move faster along their circles because v grows with r.

Common mistakes

• Mixing radians and degrees. A value of 360 means a full turn in degrees but about 57 turns in radians, so always check the angle unit.
• Using the diameter where the radius belongs. This halves or doubles the speed by mistake.
• Confusing rpm with rad/s. One rpm is only 0.1047 rad/s.
• Reading the label "Radians/m" as radians per metre. In the angular velocity list it means radians per minute, just as "Rev/h" means revolutions per hour.

For straight-line motion, see the speed calculator and the average speed calculator. To switch between speed units, use the speed converter.