Angular Acceleration Calculator – How to Calculate & Find Angular Acceleration

  • Adam
  • November 26, 2025

Calculate angular acceleration instantly with our free calculator. Learn how to find angular acceleration using formulas, see step-by-step examples, and understand instantaneous vs average angular acceleration in rad/s².

Angular Acceleration Calculator: How to Calculate Angular Acceleration

Angular acceleration is the rate at which an object's angular velocity changes over time, measured in radians per second squared (rad/s²). This calculator helps you determine angular acceleration using initial angular velocity, final angular velocity, and time duration, making it essential for analyzing rotational motion in physics and engineering applications.

Calculate Angular Acceleration

Result:

What is Angular Acceleration?

Angular acceleration (represented by the Greek letter α) measures how quickly the rotational speed of an object changes. When a spinning object speeds up or slows down its rotation, it experiences angular acceleration. The standard unit is radians per second squared (rad/s²), which indicates how many radians per second the angular velocity changes each second.

Key Characteristics of Angular Acceleration:

  • Positive angular acceleration: Occurs when rotational speed increases (acceleration in the direction of rotation)
  • Negative angular acceleration: Occurs when rotational speed decreases (deceleration or angular retardation)
  • Vector quantity: Has both magnitude and direction along the axis of rotation
  • Related to torque: Net torque on an object causes angular acceleration according to τ = Iα

Angular Acceleration Formula

The fundamental formula for calculating average angular acceleration relates the change in angular velocity to the time interval over which that change occurs:

Average Angular Acceleration:

α = (ωf - ωi) / t

α = Δω / Δt

Where:

  • α = angular acceleration (rad/s²)
  • ωf = final angular velocity (rad/s)
  • ωi = initial angular velocity (rad/s)
  • Δω = change in angular velocity (rad/s)
  • t or Δt = time interval (s)

Instantaneous Angular Acceleration

For instantaneous angular acceleration at a specific moment in time, we use calculus to find the derivative of angular velocity with respect to time:

α = dω / dt

Or equivalently:

α = d²θ / dt²

This formula represents the second derivative of angular position with respect to time, providing the instantaneous rate of change of angular velocity.

Related Kinematic Equations

Final Angular Velocity:

ωf = ωi + αt

Angular Displacement:

θ = ωit + ½αt²

Velocity-Displacement Relation:

ωf² = ωi² + 2αθ

Relationship to Linear Acceleration:

at = αr

(where at is tangential linear acceleration and r is radius)

How to Calculate Angular Acceleration

Follow these systematic steps to calculate angular acceleration accurately:

  1. Identify the known values: Determine which quantities you have—initial angular velocity (ωi), final angular velocity (ωf), time (t), or angular displacement (θ).
  2. Select the appropriate formula: Choose the kinematic equation that matches your known and unknown variables.
  3. Convert units if necessary: Ensure all measurements use consistent SI units (radians for angles, seconds for time). Convert RPM to rad/s by multiplying by 2π/60.
  4. Substitute values into the formula: Plug in the known values carefully, maintaining proper signs (positive for counterclockwise, negative for clockwise by convention).
  5. Solve for angular acceleration: Perform the mathematical operations to isolate α and calculate its value.
  6. Check the sign and magnitude: Verify that positive/negative sign makes physical sense for whether rotation is speeding up or slowing down.

Calculation Example

Problem: A wheel initially rotating at 10 rad/s accelerates to 30 rad/s in 4 seconds. Find the angular acceleration.

Solution:

Given: ωi = 10 rad/s, ωf = 30 rad/s, t = 4 s

Using formula: α = (ωf - ωi) / t

α = (30 - 10) / 4

α = 20 / 4

α = 5 rad/s²

The wheel experiences a positive angular acceleration of 5 rad/s², indicating increasing rotational speed.

How to Find Instantaneous Angular Acceleration

Finding instantaneous angular acceleration requires knowledge of how angular velocity varies with time at a specific moment:

  1. Start with angular position function: Obtain the equation θ(t) that describes angular position as a function of time.
  2. Find angular velocity: Take the first derivative of angular position with respect to time: ω(t) = dθ/dt.
  3. Find angular acceleration: Take the derivative of angular velocity with respect to time: α(t) = dω/dt = d²θ/dt².
  4. Evaluate at specific time: Substitute the desired time value into the angular acceleration function to find instantaneous angular acceleration at that moment.

Instantaneous Calculation Example

Problem: If angular position is given by θ(t) = 3t³ - 6t² + 2t (in radians), find instantaneous angular acceleration at t = 2 seconds.

Solution:

Step 1: Find angular velocity by taking the first derivative:

ω(t) = dθ/dt = 9t² - 12t + 2 rad/s

Step 2: Find angular acceleration by taking the second derivative:

α(t) = dω/dt = 18t - 12 rad/s²

Step 3: Evaluate at t = 2 seconds:

α(2) = 18(2) - 12 = 36 - 12

α(2) = 24 rad/s²

At t = 2 seconds, the instantaneous angular acceleration is 24 rad/s².

Applications of Angular Acceleration

Understanding and calculating angular acceleration is crucial across multiple scientific and engineering disciplines:

  • Mechanical Engineering: Designing gears, flywheels, turbines, and rotating machinery that must handle specific acceleration profiles
  • Automotive Industry: Analyzing wheel acceleration, engine crankshaft dynamics, and transmission systems
  • Aerospace: Calculating satellite spin rates, gyroscope performance, and aircraft maneuver capabilities
  • Sports Science: Studying angular motion in gymnastics, diving, figure skating spins, and throwing events
  • Robotics: Programming robotic arm movements and controlling rotational joints with precision
  • Physics Research: Analyzing rotational dynamics, moment of inertia experiments, and conservation of angular momentum

Frequently Asked Questions

What is the difference between angular velocity and angular acceleration?
Angular velocity measures how fast an object rotates (rad/s), while angular acceleration measures how quickly that rotational speed changes (rad/s²). Angular velocity is the rate of change of angular position, whereas angular acceleration is the rate of change of angular velocity. An object can have angular velocity but zero angular acceleration if it rotates at constant speed.
How do you convert RPM to angular acceleration?
RPM (revolutions per minute) is a measure of angular velocity, not acceleration. To use RPM values in angular acceleration calculations, first convert to rad/s by multiplying by 2π/60 (or approximately 0.10472). For example, 120 RPM = 120 × (2π/60) = 12.57 rad/s. Then use this value in the angular acceleration formula α = (ωf - ωi) / t.
Can angular acceleration be negative?
Yes, negative angular acceleration indicates that the rotational speed is decreasing (angular deceleration). By convention, counterclockwise rotation is positive and clockwise is negative. If an object rotating counterclockwise is slowing down, or an object rotating clockwise is speeding up, the angular acceleration will be negative. The sign depends on the chosen coordinate system and direction of rotation.
What is the relationship between angular and linear acceleration?
Linear tangential acceleration (at) and angular acceleration (α) are related by the equation at = αr, where r is the distance from the rotation axis. This means points farther from the axis experience greater linear acceleration for the same angular acceleration. Additionally, there is centripetal acceleration ac = ω²r directed toward the rotation center.
What causes angular acceleration in rotating objects?
Angular acceleration is caused by net torque acting on an object, as described by the equation τ = Iα, where τ is torque and I is moment of inertia. Greater torque produces greater angular acceleration, while objects with larger moment of inertia (more mass distributed farther from the axis) accelerate more slowly for the same torque. This is Newton's second law applied to rotational motion.
How is instantaneous angular acceleration different from average angular acceleration?
Average angular acceleration is calculated over a finite time interval using α = Δω/Δt, giving one value for the entire period. Instantaneous angular acceleration is the value at a specific moment in time, found using calculus (α = dω/dt). For uniform angular acceleration, both values are equal. For non-uniform acceleration, instantaneous values vary continuously while average gives a single representative value.
What units are used for angular acceleration?
The SI unit for angular acceleration is radians per second squared (rad/s²). Since radians are dimensionless (they represent a ratio of arc length to radius), the unit can also be expressed as s⁻². Other units include degrees per second squared (°/s²) or revolutions per second squared (rev/s²), though rad/s² is the standard in physics and engineering calculations.
Can you have angular acceleration with zero angular velocity?
Yes, an object can have zero angular velocity at an instant while still experiencing angular acceleration. This occurs at the turning points of oscillating rotational motion, similar to how a pendulum has zero linear velocity at its highest point but still has acceleration. The object is momentarily stationary in rotation but changing direction, so it must be accelerating.

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