Angular velocity ω measures how fast something spins, in radians per second — the rotational twin of ordinary speed. Engineers usually quote rotation as RPM, so the workhorse conversion is ω = 2πN/60, and since one turn is 2π radians, ω = 2πf links it to frequency too. Multiply by a radius and you get the linear speed v = ωr at that point — the reason the edge of a wheel races along while the hub barely moves. RPM, rad/s and hertz are simply three views of one rate.
Reviewed: June 20, 2026 · Author: Naveen P N, Founder — AI Calculator · Verified against: ω = 2πf and v = ωr.
The angular velocity equations
One revolution sweeps 2π radians, and there are 60 seconds in a minute, so RPM converts to rad/s by multiplying by 2π/60 ≈ 0.1047. Frequency in hertz is just RPM divided by 60, and ω = 2πf. To get the tangential speed at the rim, multiply ω by the radius. All three rotation units describe the same spin; the radius is what turns that spin into a real linear speed at a chosen distance from the axis.
Worked example — a turntable
Scenario: A turntable spins at 60 RPM. What is its angular velocity, frequency, and the rim speed at a 0.5 m radius?
At 60 RPM the table turns once per second — 6.283 rad/s, or 1 Hz. A point on the 0.5 m rim travels at 3.14 m/s, while a point at half that radius moves at only 1.57 m/s for the same rotation. Speed up to 1000 RPM and ω jumps to 104.7 rad/s (16.67 Hz); the rim speed scales with it. RPM is convenient for machines, but rotational physics needs ω in rad/s.
Frequently Asked Questions
ω = 2πN/60. 60 RPM → 6.283 rad/s. RPM = 60ω/(2π) the other way.
The rotation rate in rad/s — radians swept per second. RPM, Hz and rad/s are the same spin.
v = ωr. ω = 6.283 rad/s at r = 0.5 m → 3.14 m/s. Bigger radius, faster rim.
× 2π/60 ≈ 0.1047. 100 RPM ≈ 10.47 rad/s, 1000 RPM ≈ 104.7 rad/s.
ω = 2πf. Frequency counts turns/sec (Hz); ω measures radians/sec. 1 Hz = 6.283 rad/s.