# Angular Acceleration Calculator

Calculate angular acceleration, displacement, revolutions, and tangential acceleration from initial/final angular velocity and time.

## What this calculates

Angular acceleration measures how quickly an object's angular velocity changes over time. Defined as alpha = (omega_2 - omega_1) / t, it is the rotational analog of linear acceleration. This calculator also computes angular displacement, number of revolutions, and tangential linear acceleration at a given radius.

## Inputs

- **Initial Angular Velocity (ω₁)** (rad/s) — Angular velocity at the start, in radians per second.
- **Final Angular Velocity (ω₂)** (rad/s) — Angular velocity at the end, in radians per second.
- **Time** (s) — min 0.001 — Time interval over which the angular velocity changes.
- **Radius (optional)** (m) — min 0 — Radius for computing tangential (linear) acceleration. Leave 0 to skip.

## Outputs

- **Angular Acceleration (α)** (rad/s²) — Rate of change of angular velocity (α = (ω₂ - ω₁) / t).
- **Angular Displacement (θ)** (rad) — Total angle swept during the acceleration (θ = ω₁t + ½αt²).
- **Number of Revolutions** — Angular displacement converted to full revolutions.
- **Tangential Linear Acceleration** (m/s²) — Linear acceleration at the given radius (a = α × r).

## Details

Angular acceleration (α) is measured in rad/s² and describes the rate of change of angular velocity. Just as linear kinematics has equations for constant acceleration, rotational kinematics has analogous equations: α = (ω₂ - ω₁)/t, θ = ω₁t + ½αt², and ω₂² = ω₁² + 2αθ.

The angular displacement θ gives the total angle swept by the rotating object. Dividing by 2π converts to revolutions, which is useful for applications like motors, wheels, and gears. For example, a motor that sweeps 62.8 radians has completed exactly 10 revolutions.

The tangential linear acceleration at a point on the rotating body is a = α × r, where r is the distance from the rotation axis. This connects rotational and linear motion: a point on the rim of a spinning wheel experiences both tangential acceleration (from changing speed) and centripetal acceleration (from changing direction). Angular acceleration is fundamental to the design of engines, turbines, gyroscopes, and any rotating machinery.

## Frequently Asked Questions

**Q: What is the difference between angular and linear acceleration?**

A: Angular acceleration (α, in rad/s²) describes how quickly an object's rotation rate changes. Linear acceleration (a, in m/s²) describes how quickly an object's straight-line speed changes. They are related by a = α × r at a distance r from the axis.

**Q: How do I convert RPM to rad/s?**

A: Multiply RPM by 2π/60. For example, 3000 RPM = 3000 × 2π/60 = 314.16 rad/s. To convert back, multiply rad/s by 60/(2π).

**Q: What causes angular acceleration?**

A: Angular acceleration is caused by a net torque: α = τ/I, where τ is the net torque and I is the moment of inertia. This is the rotational analog of Newton's second law (F = ma).

**Q: Can angular acceleration be negative?**

A: Yes. Negative angular acceleration means the rotation is slowing down (if angular velocity is positive) or speeding up in the opposite direction. It simply means the angular velocity is decreasing algebraically.

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Source: https://vastcalc.com/calculators/physics/angular-acceleration
Category: Physics
Last updated: 2026-04-21