Angular Velocity Calculator
Angular velocity ω from revolutions per minute, a period, a frequency or a linear speed and radius (ω = v ÷ r), converted between rad/s, rpm, deg/s and Hz, or the angular acceleration from a change in ω over a time.
Result…
How it's calculated
ω = 2π × rpm ÷ 60 = 2π f = 2π ÷ T = v ÷ r 1 rad/s = 60 ÷ 2π rpm = 180 ÷ π deg/s α = (ω₂ − ω₁) ÷ t Angle turned = (ω₁ + ω₂) ÷ 2 × t
Angular velocity measures how fast something turns, in radians per second. One full turn is 2π radians, so revolutions per minute convert with ω = 2π × rpm ÷ 60, a frequency in turns per second with ω = 2π f, and a period (the time for one turn) with ω = 2π ÷ T. For a point moving at speed v on a circle of radius r, ω = v ÷ r. Angular acceleration is the change in angular velocity divided by the time it took, assuming a steady rate. Unit factors are exact: 1 mph = 0.44704 m/s, 1 ft = 0.3048 m, 1 in = 0.0254 m. For the force needed to hold something on a circle, use the centripetal force calculator.
Your numbers, step by step
About the Angular Velocity calculator
This angular velocity calculator turns whatever you know about a spinning object into ω in radians per second, and converts it to revolutions per minute, degrees per second and frequency in hertz, with the time for one turn. Start from rpm, rad/s, deg/s, a frequency, a period, or a linear speed and radius using ω = v ÷ r, with speed in m/s, km/h, mph or ft/s and radius in metres, centimetres, millimetres, feet or inches. A second mode finds the angular acceleration from a starting and final angular velocity and the time between them, plus the angle turned and the number of turns. It suits physics homework, motors, wheels, fans, turntables and drills. Every conversion is shown with your numbers, using exact unit factors.
How to use it
- Choose what to “Find”: angular velocity or angular acceleration.
- For angular velocity, pick what “I know the” value is, such as “Revolutions per minute” or “Linear speed and radius”, and enter it.
- For angular acceleration, enter the “Starting angular velocity”, the “Final angular velocity”, their unit and the “Time taken”.
- Read ω in rad/s and the rpm, deg/s, hertz and period underneath, or the acceleration and the turns made.
Worked example
A car wheel with a rolling radius of 0.32 m turns at a road speed of 100 km/h. 100 km/h is 27.78 m/s, so ω = 27.78 ÷ 0.32 = 86.81 rad/s. That is about 828.9 rpm, 13.82 turns per second, and one turn every 0.0724 s. In the second mode, a motor going from 0 to 7200 rpm in 4 s has an angular acceleration of 188.5 rad/s² (1800 rpm per second) and makes 240 turns on the way.
Updated
Questions people ask
How do I calculate angular velocity from rpm?
Multiply the rpm by 2π and divide by 60: ω = 2π × rpm ÷ 60. So 3000 rpm is 314.16 rad/s. To go back, multiply rad/s by 60 and divide by 2π.
What is the formula linking angular velocity and linear speed?
v = ω r, so ω = v ÷ r, with v in m/s, r in metres and ω in rad/s. Points further from the axis move faster, even though the whole object shares one angular velocity.
Is angular velocity in hertz the same as rad/s?
No. Hertz counts turns per second, while rad/s counts radians per second, and one turn is 2π radians. A common mistake is to put a frequency in hertz straight into a formula that needs rad/s: multiply it by 2π first.
How do I find angular acceleration?
Subtract the starting angular velocity from the final one and divide by the time: α = (ω₂ − ω₁) ÷ t, in rad/s². A negative answer means the object is slowing down.
What does this angular velocity calculator assume?
Steady rotation for ω, and a constant rate of change for the angular acceleration. It gives the size of the angular velocity, not its direction along the axis. For the inward force on something spinning, use the centripetal force calculator; for the energy stored in the spin, use the moment of inertia calculator.