Calculators
Angular Velocity Calculator
Calculate and convert angular velocity, RPM, period, and frequency.
Convert angular velocity or calculate it from rotation period, frequency, or angular displacement over time.
3.141592654 rad/s
ω = 2π/T
Period 2 s · Frequency 0.5 Hz
Angular velocity is signed when direction matters; period and frequency describe the magnitude of repeated rotation. The angle/time mode reports average angular velocity over the entered interval.
About This Tool
Angular velocity describes how quickly an angle changes with time. This calculator handles common rotation workflows in one place: use a rotation period, frequency, or angular displacement over time to calculate angular velocity, or convert an existing value between radians per second, degrees per second, revolutions per minute, and revolutions per second. It also reports the corresponding period and frequency for nonzero repeated rotation, with all calculations performed locally in your browser.
How To Use It
- Choose whether your known information is rotation period, frequency, angular displacement plus elapsed time, or an angular-velocity value to convert.
- Enter the known value and select its unit where applicable. Signed angles or angular velocities can represent opposite rotation directions.
- Choose the desired angular-velocity result unit: rad/s, deg/s, RPM, or rev/s.
- Use the reported period and frequency as magnitude-based companion values when the motion represents repeated rotation.
Examples
One revolution every 2 seconds
A 2 s period gives ω = 2π/T = π rad/s, which is 30 RPM and 0.5 Hz.
Convert 120 RPM
120 RPM equals 2 revolutions per second, so ω = 4π rad/s ≈ 12.566 rad/s and the period is 0.5 s.
From frequency
A rotating system at 5 Hz has angular speed ω = 2πf = 10π rad/s ≈ 31.416 rad/s.
Average angular velocity
A 180° angular displacement completed in 0.5 s is π radians / 0.5 s = 2π rad/s on average.
Useful Notes
Angular velocity and angular speed
Angular velocity can carry a sign or vector direction, while angular speed refers to its magnitude. In one-axis problems a positive/negative sign convention is often enough to distinguish rotation directions.
Period and frequency
For repeated uniform rotation, frequency f is the number of revolutions per second and period T is the time per revolution. They satisfy f = 1/T and the angular-speed magnitude is ω = 2πf = 2π/T.
RPM conversion
One revolution is 2π radians and one minute is 60 seconds. Therefore 1 RPM = 2π/60 rad/s, while 1 rev/s = 60 RPM = 2π rad/s.
Angle divided by time
The relationship Δθ/Δt gives average angular velocity over an interval. It equals the instantaneous angular velocity only when the rate is constant or when the interval is treated as an appropriate approximation.
Scope and assumptions
The calculator converts rotational rates and simple average motion; it does not model angular acceleration, changing speed, torque, rotational inertia, or multidimensional rotation axes. Use the appropriate dynamics equations when those effects matter.
FAQ
How do I convert RPM to rad/s?
Multiply RPM by 2π/60. For example, 60 RPM equals 2π rad/s, approximately 6.283 rad/s.
What is the difference between Hz and RPM?
Hz is cycles or revolutions per second for periodic rotation, while RPM is revolutions per minute. For rotation, 1 Hz corresponds to 60 RPM.
Can angular velocity be negative?
Yes. A negative value can represent rotation opposite to the direction defined as positive. The choice of positive direction is a convention.
Why is period unavailable at zero angular velocity?
A stopped object does not complete repeated revolutions, so a finite rotation period or nonzero rotation frequency cannot be derived from zero angular speed.
Does angle divided by time give instantaneous angular velocity?
It gives average angular velocity over that interval. Instantaneous angular velocity requires the rate of change of angle at a particular instant.
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