Angular Velocity Converter
Same value in every unit
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Why rpm has to become rad/s before the physics works
Almost every rotational formula expects angular velocity in radians per second: linear speed v = ωr, centripetal acceleration ω²r, kinetic energy ½Iω². Feed one of them a value in rpm and the answer is wrong quietly, with no error message — too large by a factor of about 9.55 wherever ω appears once, and by about 91 wherever it is squared.
The conversion is a single number. Multiply rpm by 2π/60, about 0.10472, to get rad/s. Go the other way by multiplying rad/s by 60/2π, about 9.5493. A petrol engine at 3000 rpm is turning at 100π, or about 314 rad/s.
Where 0.1047 comes from
One revolution is 2π radians and one minute is 60 seconds, so rpm to rad/s is just 2π radians per 60 seconds. Every other pairing on this page is the same bookkeeping with different units swapped in: degrees instead of radians (π/180 per degree), or minutes and hours instead of seconds. None of it depends on the size of the rotating object.
Angular velocity is not linear speed
Every point on a rigid rotating disc shares the same angular velocity. The rim and a point halfway to the centre complete a revolution in exactly the same time. Linear speed does not work that way — v = ωr, so it climbs with radius. Halve the radius and you halve the surface speed at identical rpm.
This is why bonded abrasives carry a maximum operating speed in metres per second rather than rpm alone. A 115 mm angle-grinder disc at 11,000 rpm gives ω ≈ 1152 rad/s, so the rim is doing roughly 66 m/s while a point 10 mm from the centre is under 12 m/s. As the disc wears smaller, the same rpm produces a slower cut.
Useful anchors
- 1 rpm ≈ 0.1047 rad/s; 1 rad/s ≈ 9.549 rpm
- 60 rpm = 1 revolution per second = 2π ≈ 6.283 rad/s
- 1 rad/s ≈ 57.30 degrees per second
- A clock's second hand: exactly 1 rpm, ≈ 0.1047 rad/s; its minute hand: 1 revolution per hour, ≈ 0.001745 rad/s
- Two-pole synchronous motor, 50 Hz supply: exactly 3000 rpm, ≈ 314.2 rad/s (60 Hz: 3600 rpm, ≈ 377 rad/s)
- A 7200 rpm hard disc: ≈ 754 rad/s
- The Earth: one turn per sidereal day, ≈ 7.292 × 10⁻⁵ rad/s
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Frequently Asked Questions
Multiply the rpm figure by 2π/60, which is about 0.10472. So 1500 rpm × 0.10472 ≈ 157.1 rad/s. The factor exists because one revolution is 2π radians and one minute is 60 seconds, so it applies to anything that rotates, regardless of its size or mass.
About 314 rad/s. Written out, 3000 × 2π/60 is exactly 100π, which is 314.16 to two decimal places. The figure turns up often because a two-pole synchronous motor on a 50 Hz supply runs at exactly 3000 rpm, and it sits well inside the normal working range of a petrol car engine.
Multiply by 60/2π, about 9.5493, or divide by 0.10472 — they are the same operation. So 100 rad/s ≈ 955 rpm. For a rough mental check, multiplying rad/s by 10 overshoots the true rpm figure by only about 4.7 per cent.
No. Angular velocity describes how fast something turns and is identical at every point on a rigid rotating body. Linear speed describes how fast one particular point travels through space, and it depends on radius through v = ωr. A point on the axis of a spinning disc has zero linear speed while the rim may be moving at tens of metres per second.
Hertz counts complete revolutions per second, so it is rpm divided by 60: 3000 rpm = 50 Hz. Radians per second counts the angle swept in that time, so ω = 2πf. The three get mixed up because rotational frequency and angular frequency are both loosely called frequency, yet they differ by a factor of 2π.
Because linear speed rises with radius while angular velocity stays the same across the whole wheel. On a 115 mm disc at 11,000 rpm the angular velocity is about 1152 rad/s, giving a rim speed near 66 m/s, while a point 10 mm from the centre is under 12 m/s. That is why bonded wheels are marked with a maximum operating speed in metres per second, with the rpm limit following from the diameter.