Free · Rotational and periodic motion calculators 🌎

Angular Acceleration from Rate Change or Tangential Motion

Calculate signed average angular acceleration from initial/final rotation rates and elapsed time, or from tangential acceleration and radius, with all captured units and inverses.

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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

Changing angular velocity from 1 to 5 rad/s in 2 seconds gives Δω=4 rad/s and average acceleration 2 rad/s². Tangential acceleration 6 m/s² at radius 3 m also corresponds to 2 rad/s².

α_average = (ω_final−ω_initial)/t; Δω=ω_final−ω_initial; a_t=αr.

Worked example

Enter these known values and leave the other values blank.

Initial angular velocity ω₁
1 rad/s
Final angular velocity ω₂
5 rad/s
Elapsed time t
2 sec
Signed angular velocity change Δω
4 rad/s
Average angular acceleration α
2 rad / s²

Assumptions and limitations

  • The rate-change method gives interval-average angular acceleration. Tangential acceleration divided by radius describes the matching instantaneous angular component, or a consistent average at fixed radius. The method selector preserves the two captured views of the same linked angular-acceleration quantity.
  • Choose a fixed-axis sign convention for initial/final rates, their difference, and tangential acceleration. Negative angular acceleration is valid and does not always mean slowing down: its relation to the sign of angular velocity matters.
  • Elapsed time and radius must be positive. Equal initial/final rates give zero average acceleration for a known positive interval. Zero tangential acceleration at known radius also gives zero angular acceleration.
  • A zero rate change with zero acceleration does not specify a unique time. A zero tangential/angular acceleration pair does not specify a unique radius. Those dividing inverses remain blank; enter additional information.
  • The tangential relation excludes the radial/centripetal acceleration component. This tool does not calculate total acceleration vectors, torque, inertia or a motion history between the supplied endpoints.
  • Angular acceleration uses rad/s² or deg/s² in the rate-change view and fixed rad/s² in the tangential view. The g selector uses standard gravity 9.80665 m/s². Duration years are 365.25 days and months 30.4375 days, not calendar intervals.

Common questions

Does negative acceleration always slow the rotation?

No. It changes angular velocity in the negative direction. Speed decreases only when acceleration opposes the current angular-velocity sign.

Can equal endpoint rates determine elapsed time?

With zero angular acceleration, any positive interval gives the same zero rate change. Supply the elapsed time rather than dividing zero by zero.

Is tangential acceleration the total acceleration?

No. Circular motion can also have a radial component. The a_t=αr relation describes the tangential component only.

References

The tangential method displays the alpha2 alias while its canonical alpha field is hidden. Validate the visible identity a_t=alpha2*r with the same tolerance as equation consistency, so locked incompatible visible inputs cannot evade validation through a hidden alias. 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. Zero angular acceleration and zero tangential acceleration are valid; they do not uniquely determine elapsed time or radius from a zero pair. Bookify uses the exact relationship one revolution = 2π radians instead of the captured rounded angular-speed factor.

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