Angular Velocity Calculator

angular velocity = 2π ÷ period linear speed = angular velocity × radius rotational speed in rpm = angular velocity × 60 ÷ 2π

n

Rotational speed, in rpm by default — the field also takes rad/s, °/s and rev/s. This is the number off a tachometer, a motor nameplate or a wheel, and it must be greater than zero

r

Radius from the axis of rotation to the point you care about, in metres by default — the field also takes cm, mm, km, inches and feet. Zero is allowed and gives a linear speed of zero: the axis itself does not move

ω

Angular velocity — the primary result, always printed in radians per second. This is the form the rest of mechanics needs: torque, angular momentum and centripetal force all take ω, not rpm

T

Period of one turn, in seconds — the time from one turn to the next. It is the reciprocal of the rotational speed, so twice the speed is half the period

v

Linear speed of a point at the radius you entered, in metres per second — along the circle it travels, not the straight-line distance it covers

Use this page when something spins and you want to know what its rotation means in the units the rest of the physics uses — sizing a pulley or a gear, checking a wheel's rim speed against its tyre rating, working out a fan or a grinder's surface speed, or converting a datasheet's 7,200 rpm into the rad/s that a torque or centripetal force calculation needs. Three habits make the answers more useful. First, read the period as a sanity check: one turn per minute is a period of exactly one second, so a period of 0.008 s belongs to something doing thousands of revolutions a minute — a hard disk, not a wind turbine. Second, remember the radius is doing real work — the same rotation at twice the radius is twice the linear speed, which is why a wheel's rim, not its hub, is what limits how fast a vehicle can go. Third, do not confuse this page with the angular acceleration page: this one describes a rotation that is already happening, and it cannot tell you whether the speed is rising or falling. If you need to know how fast the rotation is changing, that is a different calculation with a different input — the starting and finishing speeds and the time between them.

1. A car wheel, 800 rpm, with the tyre 0.3 m from the axle

   1. 800 rpm in revolutions per second: 800 ÷ 60 = 13.333 rev/s
   2. In rad/s: 13.333 × 2π = 83.776 rad/s
   3. Period: 2π ÷ 83.776 = 0.075 s, which is 1 ÷ 13.333 rev/s
   4. Linear speed: 83.776 × 0.3 = 25.133 m/s
   5. In km/h, since that is what a speedometer shows: 25.133 × 3.6 = 90.5 km/h

   The interesting number here is the last one, because it is the one you can check against the world: an 800 rpm wheel of that size is a car doing about 90 km/h, which is a believable motorway speed and a believable wheel size. It also shows why the radius had to be an input rather than something the page could look up — the rotation alone does not say how fast the car is going. The period is worth reading too: 0.075 s means the wheel turns thirteen times a second, and the tyre's contact patch is being flexed thirteen times a second, which is the frequency that tyre noise and tread wear live at.

2. A bench grinder, 1,400 rpm, with a 0.15 m wheel

   1. 1,400 rpm in revolutions per second: 1,400 ÷ 60 = 23.333 rev/s
   2. In rad/s: 23.333 × 2π = 146.608 rad/s
   3. Period: 2π ÷ 146.608 = 0.043 s
   4. Linear speed: 146.608 × 0.15 = 21.991 m/s
   5. That is about 22 m/s at the rim, or roughly 79 km/h

   This is the case where the rim speed is the whole point. A grinding wheel's rating is written as a maximum surface speed in metres per second, not as an rpm, because the same rpm is safe on a small wheel and dangerous on a large one — the wheel is held together by its own tensile strength, and the load on it grows with the square of the rim speed. 1,400 rpm is what a four-pole motor turns at on a 50 Hz supply, and 22 m/s sits comfortably inside a typical wheel's rating; put a 0.3 m wheel on the same spindle and the rim speed doubles to 44 m/s with no change to the motor at all.

3. A shaft turning exactly once a minute, with a point 1 m out

   1. 60 rpm is 60 revolutions in 60 seconds, which is 1 revolution per second
   2. Period: 1 ÷ 1 rev/s = 1 s — one turn per second, so one second per turn
   3. In rad/s: 1 rev/s × 2π = 6.283 rad/s, the same as 60 × 2π ÷ 60
   4. Check against the period: 2π ÷ 6.283 = 1 s, so the two agree
   5. Linear speed at 1 m: 6.283 × 1 = 6.283 m/s

   This example exists to pin down the one rotational speed whose period you can check without any arithmetic. At 60 rpm the period is exactly one second, because 60 revolutions in 60 seconds is one revolution per second — and that reference is what keeps a period of 0.008 s from being misread by a factor of ten: it is thousands of revolutions a minute, a hard disk platter, not hundreds. It also separates two numbers that get written the same way and differ by a factor of 60: 60 rpm is 1 rev/s, while 60 rev/s would be 3,600 rpm. The linear speed then falls out of the circumference, because a point at 1 m travels 2πr = 6.283 m in exactly one second. Real things at 60 rpm: a clock's second hand, a slow cement mixer, and the output of a gearbox geared well down for torque.

The page describes a rotation at a single instant — one state of motion — and says nothing about how that state was reached or how long it lasts. Nothing here is about torque, power or moment of inertia: knowing that a shaft turns at 314 rad/s does not tell you what is driving it, because the same ω takes a tiny motor on an unloaded shaft and a large one on a loaded flywheel. The linear speed is the speed of a point moving along a circle, and it is not a velocity in the direction the machine is travelling: a car's wheel rim has a linear speed of 25 m/s relative to the axle, but the car's forward speed is a separate quantity that happens to equal it only when the wheel is rolling without slipping. The radius is treated as a fixed distance, so nothing here covers a changing radius — a spinning ice skater pulling her arms in, a cable winding onto a drum, or a turbine blade that flexes. The page also assumes ideal rigidity and no slip: a belt, a tyre or a clutch that slips makes the linear speed at the output wrong by exactly the slip ratio. Finally, a rotation reversing direction is outside what this page can express — rpm is treated as a magnitude, and the fields reject negative values, so a shaft that runs backwards has to be entered as its speed with the direction noted elsewhere.

How do I calculate angular velocity from rpm?

Multiply the rpm by 2π and divide by 60, which is the same as multiplying by 0.10472. A shaft turning at 3,000 rpm therefore has an angular velocity of 3,000 × 2π ÷ 60 = 314.159 rad/s. The reason the factor is there at all is that one turn is 2π radians and one minute is 60 seconds, so converting rpm to rad/s changes both the angle unit and the time unit at once. The page does that conversion in the primary result, so you can leave the field in rpm and read the answer in radians per second.

Why is the radius an input if I only want the angular velocity?

Because the card in the tool list promises three things — angular velocity, period and linear speed — and the linear speed needs a radius. There is no way to print a linear speed without knowing the distance from the axis: a point on the axis does not move at all, and the same rotation at twice the radius is twice as fast. The radius is set to 0.1 m by default so it never blocks you, and it has no effect whatsoever on the other two outputs — change it and the rad/s and the period stay exactly where they were.

What is the difference between this page and the angular acceleration calculator?

This page describes a rotation that is going on right now: one angular velocity, one period, one linear speed. Angular acceleration is about change — how fast the rotation is speeding up or slowing down — and its formula is α = (ωf − ωi) ÷ t, which needs two angular velocities and a time. A shaft at a steady 3,000 rpm has an angular velocity and no angular acceleration at all, and a shaft that is speeding up has both. The two pages share the same unit table for rotation and nothing else, and mixing them up produces an answer that is off by a factor of the time.

How do I work out the period of one revolution?

Divide 2π by the angular velocity, or equivalently divide 60 by the rpm. At 3,000 rpm the period is 60 ÷ 3,000 = 0.02 seconds, which is the time from one turn to the next — so the shaft makes fifty turns a second. The period is often the more useful of the two when you are thinking about vibration, because the reciprocal of the period is the frequency at which the rotation repeats. A wheel doing thirteen turns a second, as in the first example, is flexing its tyre thirteen times a second, and that rate is the period inverted.

What is the linear speed of a point on a rotating object?

It is the angular velocity multiplied by the radius, v = ωr, and it is the speed along the circle rather than in any straight line. A point 0.3 m from the axle of a wheel turning at 83.776 rad/s moves at 25.133 m/s — about 90 km/h — and a point on the same wheel at 0.15 m moves at exactly half that. The axis itself has a linear speed of zero, which is why the page allows a radius of zero and reports zero rather than complaining. The one thing the number is not is the speed of the vehicle: for a wheel rolling without slipping the two are equal, and once the tyre slips they are not.

Should I use rev/s or Hz for rotational speed?

They have the same numerical value and they are not the same unit, and the page offers rev/s in the field but never reports Hz. A rotation of 3,000 rpm is 50 rev/s, and 50 Hz is also the mains frequency a four-pole motor is synchronised to, so the two numbers coincide — but one counts turns and the other counts cycles of a periodic quantity. Writing rev/s as Hz is how a rotational speed gets read as a frequency, which matters as soon as a page is talking about resonance, vibration or a filter, where the Hz is the thing being designed for and the rpm is a mechanical consequence.

• Rotational Motion and Angular Momentum (College Physics 2e, §6.1) — ω = Δθ ÷ Δt, the relation between period and angular velocity, and the definition of the radian — OpenStax
• The International System of Units (SI Brochure, 9th edition) — the radian as the coherent SI unit of plane angle, so that rad/s and s⁻¹ coincide numerically without being interchangeable names — Bureau International des Poids et Mesures
• Linear speed and the relation v = ωr — including why the radius must appear explicitly and why two points on the same rotating body have different linear speeds — Wikipedia
• ANSI B7.1 — Safety Requirements for the Use, Care and Protection of Abrasive Wheels, the standard behind the maximum surface speed marked on a grinding wheel — American National Standards Institute