Angular acceleration (symbol: α, alpha) is the rate at which an object's angular velocity changes over time. It is measured in units of angle per time squared, with the SI unit being radians per second squared (rad/s²). Mathematically, it is defined as α = Δω/Δt, where ω is angular velocity and t is time. In calculus form, α = dω/dt = d²θ/dt².
Angular Acceleration Converter
Convert angular acceleration between rad/s², deg/s², rev/s², rev/min², rad/min² and 20+ units with scientific precision. Instant results with formula explanations.
Conversion Formula
1 rad/s² = 57.295779513 deg/s²
Common Conversions
Quick reference table (from rad/s²)| From (rad/s²) | To | Result |
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What is Angular Acceleration?
Angular acceleration (symbol: α, alpha) is the rate of change of angular velocity with respect to time. It describes how quickly an object's rotational speed is changing, measured in units of angle per time squared.
In the International System of Units (SI), angular acceleration is measured in radians per second squared (rad/s²). One radian per second squared means the angular velocity increases by one radian per second every second.
Angular acceleration is the rotational equivalent of linear acceleration. Just as linear acceleration describes how quickly an object's speed changes along a straight path, angular acceleration describes how quickly an object's rotation rate changes.
Basic Formula
α = Δω / Δt
Angular acceleration = change in angular velocity ÷ time
Angular acceleration = change in angular velocity ÷ time
Calculus Form
α = dω/dt = d²θ/dt²
First derivative of angular velocity, second derivative of angular position
First derivative of angular velocity, second derivative of angular position
Torque Relation
τ = I × α
Torque = moment of inertia × angular acceleration
Torque = moment of inertia × angular acceleration
Linear Relation
aₜ = α × r
Tangential acceleration = angular acceleration × radius
Tangential acceleration = angular acceleration × radius
Angular vs Linear Acceleration
Understanding the relationship between angular and linear acceleration is fundamental to rotational dynamics. While linear acceleration describes motion along a straight path, angular acceleration describes rotational motion around an axis.
Linear Acceleration (a)
Measured in m/s². Describes how quickly velocity changes along a straight path.
Angular Acceleration (α)
Measured in rad/s². Describes how quickly angular velocity changes around an axis.
Relationship: aₜ = α × r
Tangential linear acceleration equals angular acceleration multiplied by radius.
Centripetal Acceleration
aᶜ = ω² × r — directed toward the center, perpendicular to tangential acceleration.
Total Acceleration
a = √(aₜ² + aᶜ²) — vector sum of tangential and centripetal components.
Newton's Second Law (Rotational)
τ = I × α — torque equals moment of inertia times angular acceleration.
Angular Acceleration Units Reference
Angular acceleration can be expressed in many different units depending on the application. Here is a comprehensive reference of all supported units with their conversion factors to the SI unit (rad/s²).
| Unit | Symbol | Equals (rad/s²) | Common Use |
|---|---|---|---|
| Radian per second squared | rad/s² | 1 | SI unit, physics |
| Degree per second squared | deg/s² | π/180 ≈ 0.01745 | Engineering, navigation |
| Revolution per second squared | rev/s² | 2π ≈ 6.28319 | Rotating machinery |
| Revolution per minute squared | rev/min² | 2π/3600 ≈ 0.001745 | Motors, turbines |
| Radian per minute squared | rad/min² | 1/3600 ≈ 0.000278 | Slow rotations |
| Degree per minute squared | deg/min² | π/648000 ≈ 4.85e-6 | Astronomy |
| Gradian per second squared | grad/s² | π/200 ≈ 0.01571 | Surveying |
| Gradian per minute squared | grad/min² | π/720000 ≈ 4.36e-6 | Surveying |
| Arcminute per second squared | arcmin/s² | π/10800 ≈ 0.000291 | Astronomy |
| Arcsecond per second squared | arcsec/s² | π/648000 ≈ 4.85e-6 | Telescopes |
| Sign per second squared | sign/s² | π/6 ≈ 0.52360 | Astrology |
| Angular mil per second squared | mil/s² | 2π/6400 ≈ 0.000982 | Military, artillery |
| Circle per second squared | circle/s² | 2π ≈ 6.28319 | Full rotations |
| Circle per minute squared | circle/min² | 2π/3600 ≈ 0.001745 | Full rotations |
| Turn per second squared | turn/s² | 2π ≈ 6.28319 | Engineering |
| Gon per second squared | gon/s² | π/200 ≈ 0.01571 | Surveying |
| Revolution per hour squared | rev/h² | 2π/12960000 ≈ 4.85e-7 | Very slow rotation |
| Radian per hour squared | rad/h² | 1/12960000 ≈ 7.72e-8 | Very slow rotation |
| Degree per hour squared | deg/h² | π/233280000 ≈ 1.35e-9 | Very slow rotation |
Real-World Applications
Angular acceleration is fundamental to understanding rotational motion in countless real-world applications. From the spinning of a hard drive to the rotation of planets, angular acceleration plays a crucial role.
Automotive Engineering
Engine crankshaft acceleration, wheel spin-up during acceleration, and brake deceleration calculations.
Aerospace
Aircraft roll rates, satellite attitude control, rocket spin stabilization, and gyroscope dynamics.
Electric Motors
Motor startup acceleration, load response, and speed regulation in industrial applications.
Robotics
Joint acceleration control, manipulator dynamics, and precise rotational positioning.
Hard Disk Drives
Spindle motor acceleration to operating speed (typically 5400-7200 RPM) and deceleration during shutdown.
Sports Science
Analysis of pitching mechanics, golf swing dynamics, and figure skating spin acceleration.
Astronomy
Pulsar spin-down rates, planetary rotation changes, and tidal braking effects.
Power Generation
Turbine startup acceleration, generator synchronization, and grid frequency regulation.
How to Use This Converter
Our angular acceleration converter is engineered for speed, precision, and ease of use. Follow these steps to get accurate conversions instantly.
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Choose Source Unit Pick the unit you want to convert from (e.g., rad/s², deg/s², rev/s²).
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Choose Target Unit Select the unit you want to convert to in the second dropdown.
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Enter Value Type your angular acceleration value — the result appears instantly as you type.
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Swap Units Click the swap button to instantly reverse the conversion direction.
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View Formula The exact conversion formula is displayed below for verification and learning.
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Reference Table Use the common conversions table for quick lookups of standard values.
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Frequently Asked Questions
The SI unit of angular acceleration is radians per second squared (rad/s²). One radian per second squared means the angular velocity of an object increases by one radian per second every second. Since radians are dimensionless, the dimensional formula for angular acceleration is [T⁻²] or 1/s².
To convert from rad/s² to deg/s², multiply by 180/π (approximately 57.2958). The formula is:
deg/s² = rad/s² × (180/π)
For example: 1 rad/s² = 57.2958 deg/s², and 2 rad/s² = 114.5916 deg/s². To convert in the opposite direction, multiply by π/180.
deg/s² = rad/s² × (180/π)
For example: 1 rad/s² = 57.2958 deg/s², and 2 rad/s² = 114.5916 deg/s². To convert in the opposite direction, multiply by π/180.
Angular velocity (ω) measures how fast an object is rotating, expressed in rad/s. It describes the rate of change of angular position. Angular acceleration (α) measures how quickly the angular velocity is changing, expressed in rad/s². It describes the rate of change of angular velocity. Just as velocity is the derivative of position, acceleration is the derivative of velocity.
Angular acceleration is directly related to torque through Newton's second law for rotation:
τ = I × α
Where τ is torque (N·m), I is moment of inertia (kg·m²), and α is angular acceleration (rad/s²). This means that for a given torque, an object with larger moment of inertia will have smaller angular acceleration. This is the rotational equivalent of F = m × a.
τ = I × α
Where τ is torque (N·m), I is moment of inertia (kg·m²), and α is angular acceleration (rad/s²). This means that for a given torque, an object with larger moment of inertia will have smaller angular acceleration. This is the rotational equivalent of F = m × a.
The relationship is given by:
aₜ = α × r
Where aₜ is tangential linear acceleration (m/s²), α is angular acceleration (rad/s²), and r is the radius (m). This means a point farther from the axis of rotation experiences greater linear acceleration for the same angular acceleration. Note that this is only the tangential component; there is also centripetal acceleration aᶜ = ω² × r directed toward the center.
aₜ = α × r
Where aₜ is tangential linear acceleration (m/s²), α is angular acceleration (rad/s²), and r is the radius (m). This means a point farther from the axis of rotation experiences greater linear acceleration for the same angular acceleration. Note that this is only the tangential component; there is also centripetal acceleration aᶜ = ω² × r directed toward the center.
One revolution equals 2π radians. To convert from rev/s² to rad/s², multiply by 2π (approximately 6.28319):
rad/s² = rev/s² × 2π
For example: 1 rev/s² = 6.28319 rad/s², and 0.5 rev/s² = 3.14159 rad/s².
rad/s² = rev/s² × 2π
For example: 1 rev/s² = 6.28319 rad/s², and 0.5 rev/s² = 3.14159 rad/s².
Revolution per minute squared (rev/min² or rpm²) is a unit of angular acceleration commonly used in engineering for rotating machinery like electric motors, turbines, and engines. It describes how quickly the RPM (revolutions per minute) of a rotating object is changing. To convert to rad/s²: 1 rev/min² = 2π/3600 ≈ 0.001745 rad/s².
Since radians are dimensionless (ratio of arc length to radius), the dimensional formula of angular acceleration is simply [T⁻²] or 1/s². In SI base units, it is expressed as s⁻². This is the same dimensional formula as frequency squared, though the physical meanings are entirely different.
Yes, angular acceleration can be negative. A negative angular acceleration indicates that the object is slowing down (decelerating) in its rotation, or rotating in the opposite direction. For example, when a spinning wheel is braking, its angular acceleration is negative (opposite to its angular velocity). The sign depends on the chosen coordinate system and direction of rotation.
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