Angular Velocity to Angular Acceleration Calculator

Track rotational speed changes and compute acceleration fast. Supports rad/s, deg/s, rpm conversions instantly cleanly. Download results, share links, and compare example scenarios easily.

This tool uses constant angular acceleration kinematics by default.
Angular velocity
Angular acceleration
Time
Radius (optional)
Radius enables tangential and centripetal acceleration outputs.
Signed values are allowed to represent direction.
Tip: If you solve for Δt and get a negative value, your inputs imply opposite signs. Try swapping ω₁ and ω₂, or adjust α direction.

Formula used

When angular acceleration is constant, average angular acceleration is the change in angular velocity over time:

α = (ω₂ − ω₁) / Δt

Rearrangements used by the other solve modes:

  • ω₂ = ω₁ + αΔt
  • ω₁ = ω₂ − αΔt
  • Δt = (ω₂ − ω₁) / α

The angle turned assumes constant α: θ = ω₁Δt + ½α(Δt)². If you enter a radius, the tool also shows tangential acceleration aₜ = αr and centripetal acceleration a_c = ω²r.

How to use this calculator

  1. Select what you want to solve for (α, ω₁, ω₂, or Δt).
  2. Pick units for velocity, acceleration, time, and optional radius.
  3. Fill in the required fields for the chosen solve mode.
  4. Click Calculate to view a full results table.
  5. Download CSV or PDF to save your results.

Example data table

Use the Load buttons to autofill inputs. Each row assumes constant angular acceleration.

Example Load ω₁ ω₂ Δt Computed α Notes
1 1200 rpm 1800 rpm 4 s 150 rpm/s Common motor spin-up case.
2 30 rad/s 10 rad/s 5 s -4 rad/s² Negative α indicates slowing down.
3 90 deg/s 210 deg/s 3 s 40 deg/s² Useful for turntable ramps.
4 -600 rpm 0 rpm 2 s 300 rpm/s Direction change toward rest.
5 0 rpm 3000 rpm 6 s 500 rpm/s Quick estimate for machine run-up time.
6 15 rad/s 25 rad/s 0.75 s 13.333 rad/s² Short interval acceleration check.
If your system has varying acceleration, try breaking the motion into shorter time segments and compute α for each segment.

Practical notes and examples

1) Turning angular velocity into angular acceleration

When rotation changes at a steady rate, angular acceleration is the slope of the speed change. This calculator uses α = (ω₂ − ω₁) / Δt, so a 600 rpm rise in 4 s equals 150 rpm/s, or about 15.708 rad/s².

If ω₁ is unknown, assume zero for spin‑up. Example: 3000 rpm in 6 s → 500 rpm/s, an average value.

2) Unit ranges you will see in real systems

Small fans often run near 1000–3000 rpm, while grinders may exceed 10,000 rpm. Industrial servos frequently report in rad/s and rad/s²; for reference, 1 rpm ≈ 0.10472 rad/s and 1 rad/s ≈ 9.5493 rpm.

Many motion specs use degrees per second. Since 360° equals 2π radians, 1 deg/s ≈ 0.01745 rad/s, and the same factor converts deg/s² to rad/s².

3) Spin-up time as a design target

Suppose a motor must reach 1800 rpm from rest in 2 s. The required average acceleration is 900 rpm/s, which is ≈ 94.248 rad/s². If your controller limits torque, you can increase allowed time or reduce the final speed to stay inside limits. A 0–0.5 s spin‑up is aggressive, while 0–5 s is usually manageable for most drives.

4) Relating rotation to linear edge motion

If you enter a radius, the tool also computes tangential speed v = ωr and tangential acceleration at = αr. A wheel at 25 rad/s with r = 0.30 m has v = 7.5 m/s; if α = 13.33 rad/s², then at ≈ 4.0 m/s² at the rim.

Centripetal acceleration can dominate loads: ac = ω²r. At 25 rad/s and 0.30 m, ac = 187.5 m/s².

5) Deceleration and negative values

When ω₂ is smaller than ω₁, α becomes negative. A drop from 30 rad/s to 10 rad/s in 5 s gives α = −4 rad/s². This sign matters for braking calculations and for verifying that your sign convention matches sensor orientation.

6) Working with sampled measurements

Encoders and tachometers produce discrete readings, so use times that match your sample interval. If you log every 0.05 s, compute Δt using those timestamps rather than rounded values. Averaging several adjacent samples can reduce noise before you compute α.

For sharper estimates, use a central difference: α ≈ (ω(t+Δt) − ω(t−Δt)) / (2Δt) with a small Δt.

7) Common sources of error to watch

Mixing degrees and radians is the most frequent mistake, followed by confusing rpm/s with rpm/min. Always confirm unit selectors before calculating. Also remember: α from this formula is an average over Δt, and non‑constant acceleration requires smaller intervals or a curve fit.

Very small Δt amplifies noise, while large Δt hides peaks.

FAQs

1) Can this calculator handle non‑constant acceleration?

No. It returns the average α over the chosen interval. For changing acceleration, compute results over smaller time windows or fit ω(t) with a curve, then differentiate that curve.

2) How do I convert rpm to rad/s?

Use ω(rad/s) = rpm × 2π / 60. For example, 1200 rpm × 2π/60 ≈ 125.66 rad/s. The unit selector applies these conversions automatically.

3) What does a negative angular acceleration mean?

It means the rotation rate is decreasing in your chosen sign direction. If your sensor axis is reversed, the sign may flip. The magnitude still represents how quickly the speed changes.

4) Why do I need a radius?

Radius is optional. It lets you translate rotation into rim motion: v = ωr and at = αr. This is useful for wheels, pulleys, turntables, and belt drives.

5) What is the difference between α and at?

α is angular acceleration (rotational). at is tangential linear acceleration at a radius r. They are linked by at = αr, so doubling r doubles at.

6) Which inputs are required in each solve mode?

To solve α, enter ω₁, ω₂, and Δt. To solve ω₂, enter ω₁, α, and Δt. To solve ω₁, enter ω₂, α, and Δt. To solve Δt, enter ω₁, ω₂, and α.

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Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.