# What is the angular acceleration?

Angular acceleration calculator: α = Δω ÷ Δt from two angular speeds and a time, or from the angle turned, torque and moment of inertia (α = τ ÷ I), or tangential acceleration (α = a ÷ r).

- Page: https://www.acalculator.org/physics/angular-acceleration-calculator
- JSON spec: https://www.acalculator.org/physics/angular-acceleration-calculator.json
- Version: 080a32476f8f

## Default answer

Example with the default inputs (From Speeds and time, Starting angular speed (ω₀) 0, Final angular speed (ω) 250, Angular speed unit rpm (rev/min), Time (Δt) 5 s): The angular acceleration is 5.23599 rad/s².

## Inputs

| Key | Label | Description |
| --- | --- | --- |
| by | From | What you know about the spin. |
| w0 | Starting angular speed (ω₀) | How fast it turns at the start; 0 from rest. |
| w | Final angular speed (ω) | How fast it turns at the end. |
| wu | Angular speed unit | The unit of both angular speeds. |
| t | Time (Δt) | How long the change in speed takes. |
| th | Angle turned (Δθ) | How far it turns while the speed changes, in the unit chosen below. |
| thu | Angle unit | The unit of the angle turned. |
| tq | Net torque (τ) | The sum of the torques on the body about its axis; negative slows a forward spin. |
| tqu | Torque unit | The unit of the torque. |
| I | Moment of inertia (I) | The rotational inertia of the body about the same axis. |
| Iu | Inertia unit | The unit of the moment of inertia. |
| at | Tangential acceleration (a_t) | How fast a point on the rim speeds up along its path. |
| atu | Acceleration unit | The unit of the tangential acceleration. |
| r | Radius (r) | The distance from the axis to the point. |

## Outputs

| Key | Label | Description |
| --- | --- | --- |
| result | Answer | The angular acceleration to 6 significant figures, in rad/s² and °/s². |
| alpha | Angular acceleration (rad/s²) | α in radians per second squared; negative means slowing a forward spin. |
| degS2 | Angular acceleration (°/s²) | α in degrees per second squared. |
| revS2 | Angular acceleration (rev/s²) | α in turns per second squared. |
| rpmS | Angular acceleration (rpm/s) | How many rpm the speed changes each second. |
| turns | Angle turned (rev) | Δθ = (ω₀ + ω) ÷ 2 × Δt, in turns. |
| radians | Angle turned (rad) | The same angle in radians. |
| seconds | Time taken (s) | Δt = 2Δθ ÷ (ω₀ + ω), in seconds. |

## Method

α = (ω − ω₀) ÷ Δt; α = (ω² − ω₀²) ÷ (2Δθ) with Δt = 2Δθ ÷ (ω₀ + ω); α = τ ÷ I; α = a_t ÷ r; 1 rpm = 2π ÷ 60 rad/s.

## Assumptions

- The angular acceleration is constant over the time or angle (the average for a change that is not steady).
- A rigid body turning about a fixed axis; the torque and the moment of inertia are about that same axis.

## Worked examples

1. by = speeds, w0 = 0, w = 250, wu = rpm, t = 5 gives alpha = 5.235988, degS2 = 300, rpmS = 50, turns = 10.416667, result = α = 5.23599 rad/s² (300°/s²). Source: OpenStax, University Physics Volume 1, §10.1 Rotational Variables (Equations 10.6 and 10.8; Example 10.2: 0 to 250 rpm in 5.00 s is 5.24 rad/s²), https://openstax.org/books/university-physics-volume-1/pages/10-1-rotational-variables (retrieved 2026-10-19).
2. by = speeds, w0 = 26.2, w = 0, wu = rads, t = 0.3 gives alpha = -87.333333, degS2 = -5,003.831411. Source: OpenStax, University Physics Volume 1, §10.1 Rotational Variables (Equations 10.6 and 10.8; Example 10.2: 0 to 250 rpm in 5.00 s is 5.24 rad/s²), https://openstax.org/books/university-physics-volume-1/pages/10-1-rotational-variables (retrieved 2026-10-19), Example 10.2(b) (α = −87.3 rad/s² stops 26.2 rad/s in 0.300 s).
3. by = angle, w0 = 0, w = 220, wu = rads, th = 220, thu = rad gives alpha = 110, seconds = 2. Source: OpenStax, University Physics Volume 1, §10.2 Rotation with Constant Angular Acceleration (Equations 10.11 to 10.13; Example 10.4: a reel at 110 rad/s² reaches 220 rad/s after 220 rad and 2.00 s), https://openstax.org/books/university-physics-volume-1/pages/10-2-rotation-with-constant-angular-acceleration (retrieved 2026-10-19).
4. by = torque, tq = 375, tqu = Nm, I = 56.25, Iu = kgm2 gives alpha = 6.666667. Source: OpenStax, University Physics Volume 1, §10.7 Newton’s Second Law for Rotation (Equation 10.25, Στ = Iα; Example 10.16: 375 N·m on 56.25 kg·m² gives 6.67 rad/s²), https://openstax.org/books/university-physics-volume-1/pages/10-7-newtons-second-law-for-rotation (retrieved 2026-10-19).
5. by = tangential, at = 4.95, atu = ms2, r = 0.045 gives alpha = 110. Source: OpenStax, University Physics Volume 1, §10.1 Rotational Variables (Equations 10.6 and 10.8; Example 10.2: 0 to 250 rpm in 5.00 s is 5.24 rad/s²), https://openstax.org/books/university-physics-volume-1/pages/10-1-rotational-variables (retrieved 2026-10-19), Equation 10.8 (a_t = rα), with OpenStax, University Physics Volume 1, §10.2 Rotation with Constant Angular Acceleration (Equations 10.11 to 10.13; Example 10.4: a reel at 110 rad/s² reaches 220 rad/s after 220 rad and 2.00 s), https://openstax.org/books/university-physics-volume-1/pages/10-2-rotation-with-constant-angular-acceleration (retrieved 2026-10-19): 4.95 ÷ 0.045 = 110 rad/s².

## FAQ

### How do you calculate angular acceleration?

Divide the change in angular speed by the time it takes: α = (ω − ω₀) ÷ Δt. A wheel spun from rest to 250 rpm (26.18 rad/s) in 5.00 s has α = 26.18 ÷ 5 = 5.24 rad/s².

### How do you convert rpm to rad/s?

Multiply by 2π and divide by 60: one turn is 2π radians and one minute is 60 seconds. 250 rpm = 250 × 2π ÷ 60 = 26.18 rad/s. In the same way, an angular acceleration of 1 rpm per second is 0.10472 rad/s².

### How do you find angular acceleration from torque?

Use Newton’s second law for rotation, α = τ ÷ I. A 375 N·m torque on a merry-go-round with I = 56.25 kg·m² gives α = 375 ÷ 56.25 = 6.67 rad/s².

### How do you find angular acceleration without the time?

Use the angle turned: ω² = ω₀² + 2αΔθ, so α = (ω² − ω₀²) ÷ (2Δθ). A reel that reaches 220 rad/s from rest within 220 rad has α = 220² ÷ (2 × 220) = 110 rad/s², and the time is 2Δθ ÷ (ω₀ + ω) = 2 s.

### What is the relation between angular and tangential acceleration?

A point at radius r speeds up along its path at a_t = rα, so α = a_t ÷ r. The rim of a 4.5 cm reel speeding up at 4.95 m/s² gives α = 4.95 ÷ 0.045 = 110 rad/s².

### What does a negative angular acceleration mean?

The spin is slowing down (or speeding up in the other direction). A wheel braked from 26.2 rad/s to rest in 0.300 s has α = −26.2 ÷ 0.3 = −87.3 rad/s².

## Sources

- OpenStax, University Physics Volume 1, §10.1 Rotational Variables: Equation 10.6 (α = Δω ÷ Δt), Equation 10.8 (a_t = rα); Example 10.2 (0 to 250 rpm in 5.00 s: 26.2 rad/s and 5.24 rad/s²; braking at −87.3 rad/s² stops it in 0.300 s), retrieved 2026-10-19. https://openstax.org/books/university-physics-volume-1/pages/10-1-rotational-variables
- OpenStax, University Physics Volume 1, §10.2 Rotation with Constant Angular Acceleration: Equations 10.9 to 10.13; Example 10.4 (a reel at 110 rad/s² reaches 220 rad/s in 2.00 s and 220 rad), retrieved 2026-10-19. https://openstax.org/books/university-physics-volume-1/pages/10-2-rotation-with-constant-angular-acceleration
- OpenStax, University Physics Volume 1, §10.7 Newton’s Second Law for Rotation: Equation 10.25 (Στ = Iα); Example 10.16 (375 N·m on I = 56.25 kg·m² gives 6.67 rad/s²), retrieved 2026-10-19. https://openstax.org/books/university-physics-volume-1/pages/10-7-newtons-second-law-for-rotation
- NIST Special Publication 811 (2008), Appendix B.8: 1 ft = 0.3048 m, 1 in = 0.0254 m, 1 lb = 0.45359237 kg, 1 lbf = 4.4482216152605 N, standard gravity 9.80665 m/s², retrieved 2026-10-19. https://www.nist.gov/pml/special-publication-811