Free · Rotational and periodic motion calculators 🌎

Angular Rate, Cycle Frequency & Period

Calculate mean angular rate from displacement and time, or cyclic angular frequency from cycle frequency and period.

  • Formula & worked example
  • Private in your browser
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Calculator inputs

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How to use this calculator

  1. Enter the known values in the units shown. Results update as you type.
  2. Where results are editable, change one to solve backwards. Lock a value to hold it fixed.
  3. Use the worked example to check the method. Reset restores the starting fields.

Private by default

Inputs and results stay in this browser tab. Bookify does not upload or store the values you enter.

Formula and method

An angular displacement of 6 radians over 2 seconds gives mean angular rate 3 rad/s. Separately, frequency 1 Hz corresponds to period 1 second and cyclic angular frequency 2π rad/s.

Mean angular rate = angular displacement / elapsed time; cyclic angular frequency = 2πf; period = 1/f

Worked example

Enter these known values and leave the other values blank.

Signed angular displacement
6 rad
Elapsed time
2 sec
Cycles per second
1 Hz
Mean angular rate from displacement
3 rad/s
Cycle period
1 sec
Angular frequency from cycles
6.2832 rad/s

Assumptions and limitations

  • Elapsed time, cycle period and cycle frequency are positive. Signed displacement gives a signed mean angular rate.
  • The displacement/time and frequency/period sections are separate calculations. They coincide for a uniform motion when you supply consistent cycle and displacement data.
  • One turn is exactly 2π radians. A frequency of 1 Hz corresponds to 60 revolutions per minute or 2π radians per second.

Common questions

Why are there two angular results?

One is average angular displacement divided by elapsed time. The other is the angular frequency associated with a supplied cyclic frequency. The calculator does not automatically equate the two.

Does displacement divided by time give instantaneous speed?

It gives a mean rate over the entered interval. It equals instantaneous angular velocity only for uniform rotation or in an appropriate limiting interval.

References

Bookify uses the exact relationship one revolution = 2π radians instead of the captured rounded angular-speed factor.

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