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Angular Velocity from Angle, Time or Tangential Motion

Find signed angular velocity from angle change over elapsed time or from tangential velocity and radius, with optional endpoint angles, RPM/radians/hertz and inverse controls.

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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 unwrapped 180° change in 2 seconds gives average angular velocity π/2 ≈ 1.5708 rad/s, or 15 RPM. Separately, 6 m/s at a 3 m radius gives 2 rad/s.

ω_average = Δθ/t; Δθ=θ_final−θ_initial; ω_tangent=v_t/r.

Worked example

Enter these known values and leave the other values blank.

Signed angle change Δθ
180 deg
Elapsed time t
2 sec
Signed tangential velocity v_t
6 m/s
Radius r
3 m
Average angular velocity Δθ/t
1.5708 rad/s
Angular velocity from v_t/r
2 rad/s

Assumptions and limitations

  • The two source workflows are independent. Δθ/t is an interval average; v_t/r uses tangential motion at the supplied radius. They need not agree unless they describe the same consistent constant-rotation state.
  • Angles are signed and unwrapped about a fixed axis. Endpoint subtraction does not choose the shortest path or automatically add full revolutions. Include completed turns in your angle entries.
  • Tangential velocity is the signed component along the circular path, with a positive radius. Match its sign to your angular reference. This is a scalar fixed-axis relation, not a general three-dimensional velocity vector calculation.
  • The retained source domain requires positive elapsed time/radius and nonzero angular velocity in each completed workflow. Static zero-rate states are excluded here; clear the intended unknown before an inverse.
  • One turn is 2π radians; Hz in the rotation selector means cycles per second and RPM means revolutions per minute. Their signs follow the chosen rotation reference.
  • Duration years mean 365.25 days and months one twelfth of that year (30.4375 days). These are fixed duration conversions, not calendar dates. The hours/minutes/seconds compound control preserves elapsed duration.

Common questions

Why can the two angular velocities differ?

One uses an angle change over a time interval; the other uses tangential velocity at a radius. They are independent methods and only agree when their supplied states and timing are consistent.

Do endpoint angles use the shortest rotation?

No. The tool subtracts unwrapped endpoint angles. Add complete revolutions explicitly when the motion crosses a turn boundary.

Is hertz the same numeric value as radians per second?

No. A complete cycle has 2π radians, so one rotation per second corresponds to 2π rad/s and 60 RPM.

References

Four dividing inverses require a valid positive time/radius or a nonzero angular quantity when it is their denominator. Skip singular inverses without consuming known inputs; the retained signed forward relations and all linear differences remain. The captured velocity workflows are independent estimates: angle change/time and tangential velocity/radius are not equated. Bookify uses the exact relationship one revolution = 2π radians instead of the captured rounded angular-speed factor.

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