Angular Acceleration
Angular acceleration is the rate of change of angular velocity with respect to time. Like its linear counterpart, it is a vector quantity and represents how quickly a rotating object is speeding up or slowing down its rotation. Angular acceleration is produced by a net torque and is related to it by the rotational analogue of Newton's second law.
Key Formula
α = Δω / Δt = τ / I
LaTeX: \alpha = \frac{\Delta\omega}{\Delta t} = \frac{\tau}{I}
| Symbol | Meaning | Unit |
|---|---|---|
| α | Angular acceleration | Radian per second squared (rad/s²) |
| Δω | Change in angular velocity | rad/s |
| Δt | Time interval | Second (s) |
| τ | Net torque | Newton-metre (N·m) |
| I | Moment of inertia | kg·m² |
Worked Example
Problem
A flywheel with a moment of inertia of 5 kg·m² is acted on by a net torque of 20 N·m. It starts from rest. Find the angular acceleration and the angular velocity after 4 seconds.
Solution
Step 1: Calculate angular acceleration. α = τ / I = 20 / 5 = 4 rad/s² Step 2: Calculate angular velocity after 4 s (starting from rest, ω₀ = 0). ω = ω₀ + αt = 0 + 4 × 4 = 16 rad/s
Answer
Angular acceleration α = 4 rad/s²; ω after 4 s = 16 rad/s
Comparison of Linear and Angular Kinematic Quantities
| Linear Quantity | Symbol | Angular Analogue | Symbol | Unit |
|---|---|---|---|---|
| Displacement | x | Angular displacement | θ | rad |
| Velocity | v | Angular velocity | ω | rad/s |
| Acceleration | a | Angular acceleration | α | rad/s² |
| Force | F | Torque | τ | N·m |
| Mass | m | Moment of inertia | I | kg·m² |
| Newton's 2nd law | F=ma | Rotational 2nd law | τ=Iα | — |
Interactive Tools
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Related Terms
Angular Velocity
Angular velocity is the rate of change of angular displacement of a rotating object with respect to time. It is a vector quantity whose direction is given by the right-hand rule along the axis of rotation. Angular velocity is the rotational analogue of linear velocity and is central to the analysis of rotating machinery, celestial bodies, and rigid body dynamics.
Torque
Torque is the rotational equivalent of force — it is the tendency of a force to cause rotation about a pivot or axis. Mathematically, it is the cross product of the position vector (from the axis to the point of force application) and the force vector. Torque is essential in engineering design of engines, gears, wrenches, and any rotating machinery.
Moment of Inertia
Moment of inertia is the rotational analogue of mass — it measures an object's resistance to changes in its rotational motion about a given axis. It depends on both the total mass of the object and how that mass is distributed relative to the rotation axis; mass farther from the axis contributes more. Moment of inertia is fundamental in the design of flywheels, spinning tops, gyroscopes, and all rotating mechanical systems.
From Greek "angulos" (angle) and Latin "accelerare" meaning "to hasten", from "ad-" (to) + "celer" (swift). The symbol α is the Greek letter alpha, conventionally used for angular acceleration in rotational dynamics.