Angular Velocity Calculator

How it works

From rpm: ω = 2π·rpm / 60 · From linear speed: ω = v / r · From an angle: ω = Δθ / t · From the period: ω = 2π / T · f = ω / 2π · v = ω·r · a = ω²·r · F = m·ω²·r · L = I·ω · KE = ½·I·ω² (point mass I = m·r²)

Angular velocity ω measures how fast something rotates, in radians per second. One full turn is 2π radians, so a rate in rpm converts to rad/s by multiplying by 2π and dividing by 60; a rate in revolutions per second is the frequency in hertz, and the period is one over that. You can also start from a point’s linear speed and its distance from the axis (ω = v / r), from an angle swept in a measured time, or from how long one turn takes. Every point on a rigid spinning body shares the same ω, but its linear speed grows with distance from the axis (v = ω·r), and so does the centripetal acceleration that keeps it on its circle (a = ω²·r). Enter a mass under More options to get the centripetal force, angular momentum and rotational kinetic energy, treated as a point mass at the radius.

Worked example

60 rpm is one turn per second, so ω = 2π × 60 ÷ 60 = 2π ≈ 6.2832 rad/s, 1 Hz and 360 °/s. At a 0.5 m radius the rim moves at 6.2832 × 0.5 ≈ 3.142 m/s (11.3 km/h) and feels 6.2832² × 0.5 ≈ 19.74 m/s², about 2 g. The Earth turns once in 23.934 hours, so ω = 2π ÷ 86,164 s ≈ 7.292 × 10⁻⁵ rad/s, and a point on the equator (6,371 km) moves at about 464.6 m/s.

Sources: Angular velocity — Wikipedia · NIST SI Brochure: the radian and units of angular velocity

Frequently asked questions