Angular Acceleration Calculator

Free Online Angular Acceleration Calculator (α = (ωf − ωi) ÷ t)

This angular acceleration calculator instantly solves for angular acceleration, final angular velocity, or time using the standard rotational-kinematics formula, α = (ωf − ωi) ÷ t. Enter any combination of known values — initial and final angular velocity with time, or initial angular velocity with angular acceleration and time — and the calculator converts every unit automatically, works through the full step-by-step solution, and draws a labeled before-and-after spinning-disc diagram so the entire calculation is visible at a glance.

Whether you're a physics student solving a rotational-motion word problem, an engineer analyzing how quickly a motor spins up or a flywheel brakes, or simply curious how fast a fan or wheel accelerates, this tool works in radians/second, degrees/second, revolutions/second, and RPM for angular velocity; seconds, minutes, and hours for time; and radians/second², degrees/second², revolutions/second², RPM/second, or RPM/minute for angular acceleration — and it always shows the complete working, not just a final number.

What Is Angular Acceleration? (Definition and Formula)

Angular acceleration, written with the Greek letter alpha (α), measures how quickly an object's angular velocity is changing — in other words, how fast its rate of rotation is speeding up or slowing down. Just as linear acceleration describes a change in straight-line speed, angular acceleration describes a change in rotational speed over time.

The formula for angular acceleration is one of the foundational equations of rotational kinematics:

• Angular Acceleration (α) = (Final Angular Velocity (ωf) − Initial Angular Velocity (ωi)) ÷ Time (t)
• ωi is the angular velocity at the start of the interval, ωf is the angular velocity at the end, and t is the time it took for that change to happen.
• The SI unit of angular acceleration is radians per second squared (rad/s²) — the rate of rotation increasing by 1 rad/s every second.
• Example: a fan spins up from rest (0 rad/s) to 20 rad/s in 5 seconds → Angular Acceleration = (20 − 0) ÷ 5 = 4 rad/s².

Positive vs. Negative Angular Acceleration

Angular acceleration can be positive or negative depending on whether rotation is speeding up or slowing down. A positive α means the object's spin rate is increasing — like a motor spinning up or a car wheel accelerating from a stop. A negative α (sometimes called angular deceleration) means the rotation is slowing down — like a spinning top winding down due to friction, or a grinder wheel braking to a stop. This calculator handles both automatically: simply enter ωf smaller than ωi (or a negative α directly), and the sign of the result shows the direction of change.

Example: a grinding wheel spinning at 200 rad/s comes to a stop in 2 seconds → Angular Acceleration = (0 − 200) ÷ 2 = −100 rad/s², the negative sign confirming it's decelerating.

How to Find Final Angular Velocity from Angular Acceleration and Time

If you know an object's starting angular velocity, its angular acceleration, and how long that acceleration is applied, you can rearrange the formula to find the angular velocity it reaches.

• Final Angular Velocity (ωf) = Initial Angular Velocity (ωi) + Angular Acceleration (α) × Time (t)
• Example: a wheel starts at 5 rad/s and accelerates at 2 rad/s² for 3 seconds → ωf = 5 + (2 × 3) = 11 rad/s.
• This rearrangement comes directly from multiplying both sides of α = (ωf − ωi) ÷ t by t and adding ωi.

How to Find Time from Angular Velocities and Angular Acceleration

If you know the starting and ending angular velocities along with the angular acceleration, you can solve for how long that change took.

• Time (t) = (Final Angular Velocity (ωf) − Initial Angular Velocity (ωi)) ÷ Angular Acceleration (α)
• Example: a motor needs to go from 0 to 50 rad/s at an angular acceleration of 5 rad/s² → Time = (50 − 0) ÷ 5 = 10 seconds.
• This rearrangement comes from dividing both sides of α = (ωf − ωi) ÷ t by α.

The Before/After Spinning-Disc Diagram Explained

The diagram above draws the exact scenario described by your numbers as two side-by-side spinning discs: the gray disc on the left represents the starting rotation rate ωi, and the amber disc on the right represents the ending rate ωf — with the arc length around each disc visually reflecting how fast it's spinning. The dashed amber arrow between them shows the angular acceleration α acting over time t, and if you provide a radius, the resulting tangential (rim) acceleration is displayed below in red. Every number from your inputs is labeled directly on the diagram, so the relationship between the two rotation states and the acceleration connecting them is visible at a glance.

Angular Acceleration and Tangential (Linear) Acceleration

Just as angular velocity relates to tangential velocity through the radius, angular acceleration relates to tangential (linear) acceleration the same way — every point on a rotating object shares the same angular acceleration, but points farther from the pivot experience greater linear acceleration.

• Tangential Acceleration (aₜ) = Angular Acceleration (α) × Radius (r)
• Example: a wheel with a 0.2 m radius accelerating at 4 rad/s² has a rim (tangential) acceleration of aₜ = 4 × 0.2 = 0.8 m/s².
• This calculator includes an optional radius field — enter it and the tangential acceleration is calculated automatically alongside the angular acceleration.

Common Angular Velocity, Time, and Angular Acceleration Units

Angular velocity, time, and angular acceleration can each be expressed in several unit systems depending on the field — physics coursework typically uses radians, while motor and machinery specs often use RPM. This calculator supports the units most commonly seen in each context.

• Angular velocity: radians/second (rad/s), degrees/second (°/s), revolutions/second (rev/s), revolutions/minute (RPM), radians/minute (rad/min)
• Time: seconds (s), minutes (min), hours (hr)
• Angular acceleration: radians/second² (rad/s²), degrees/second² (°/s²), revolutions/second² (rev/s²), RPM per second (RPM/s), RPM per minute (RPM/min)
• 1 rad/s² ≈ 57.2958 °/s² ≈ 9.5493 RPM/s
• 1 RPM/s = 2π ÷ 60 rad/s² ≈ 0.10472 rad/s²

How to Use This Angular Acceleration Calculator

Choose what you want to find — Angular Acceleration, Final Angular Velocity, or Time — from the dropdown at the top. Fill in the initial angular velocity along with whichever other two values you already know, in whatever units match your problem, and optionally enter a radius to also see the resulting tangential acceleration. The calculator instantly converts everything to a common base unit, applies the angular acceleration formula, and shows the result in your chosen unit — plus automatically converted into rad/s², RPM/s, and degrees/second² for quick reference.

Scroll down to the step-by-step solution for the complete working shown exactly the way you'd solve it by hand: the formula first, then your real values substituted in with unit conversion made explicit, then the arithmetic carried through to the final answer.

Worked Examples

Finding angular acceleration: a fan spins up from rest to 20 rad/s in 5 seconds → Angular Acceleration = (20 − 0) ÷ 5 = 4 rad/s².

Finding angular acceleration (deceleration): a wheel slows from 30 rad/s to 10 rad/s in 4 seconds → Angular Acceleration = (10 − 30) ÷ 4 = −5 rad/s².

Finding angular acceleration from RPM: a motor spins up from 0 to 3000 RPM in 6 seconds → converting 3000 RPM ≈ 314.16 rad/s gives Angular Acceleration ≈ 314.16 ÷ 6 ≈ 52.36 rad/s².

Finding final angular velocity: a wheel starts at 5 rad/s and accelerates at 2 rad/s² for 3 seconds → ωf = 5 + (2 × 3) = 11 rad/s.

Finding time: a motor accelerates from 0 to 50 rad/s at 5 rad/s² → Time = (50 − 0) ÷ 5 = 10 seconds.

Tangential acceleration example: a grinding wheel with a 0.05 m radius braking at −100 rad/s² has a rim deceleration of aₜ = −100 × 0.05 = −5 m/s².

Angular Acceleration vs. Linear Acceleration

Linear acceleration (measured in m/s²) describes how quickly straight-line speed changes, following a = Δv ÷ t. Angular acceleration (measured in rad/s²) describes how quickly rotational speed changes, following α = Δω ÷ t. The two are directly connected for any point on a rotating rigid body through the radius: tangential acceleration equals angular acceleration times radius (aₜ = α × r). This mirrors the same relationship between angular velocity and tangential velocity (v = ω × r), and is why doubling the radius doubles the tangential acceleration for the same angular acceleration.

Real-World Applications of Angular Acceleration

The angular acceleration formula is widely used across mechanical and electrical engineering, as well as everyday machinery:

• Electric motors and turbines — spin-up time and torque ratings are calculated directly from angular acceleration requirements.
• Automotive engineering — analyzing how quickly a wheel, turbocharger, or engine crankshaft accelerates or decelerates.
• Braking systems — calculating the angular deceleration needed for a flywheel, disc brake, or grinding wheel to stop safely within a target time.
• Robotics — controlling joint and servo motor acceleration profiles for smooth, precise rotational movement.
• Sports and biomechanics — analyzing the angular acceleration of a swinging limb, bat, or spinning gymnast during a routine.

Angular Acceleration and Torque

Angular acceleration is the rotational counterpart to Newton's second law of motion. Just as force equals mass times linear acceleration (F = m × a), torque equals moment of inertia times angular acceleration (τ = I × α), where I is the moment of inertia — the rotational equivalent of mass, describing how resistant an object is to a change in its spin rate. A larger torque produces a larger angular acceleration for the same moment of inertia, and a larger moment of inertia (more mass distributed farther from the axis) means more torque is needed to achieve the same angular acceleration. This is why flywheels — designed with a large moment of inertia — resist rapid changes in rotation speed.

Tips for Solving Angular Acceleration Word Problems

Most angular acceleration problems in physics courses and engineering specs follow a predictable pattern, and a few habits make them much easier to solve correctly:

• Identify which three of the four quantities (ωi, ωf, t, α) are given, and which one is being asked for — that tells you whether to use α = (ωf − ωi) ÷ t, ωf = ωi + α × t, or t = (ωf − ωi) ÷ α.
• Pay close attention to the sign: if the object is slowing down, ωf will be smaller than ωi, and the resulting angular acceleration will be negative — this is expected and correct, not an error.
• Convert every angular velocity and acceleration value to a consistent unit system (typically rad/s and rad/s²) before doing any arithmetic, especially when RPM values are involved.
• Sanity-check your final answer: small household motors and fans typically have angular accelerations in the range of a few rad/s² to a few tens of rad/s²; industrial motors and braking systems can reach into the hundreds. A wildly out-of-range answer usually means a unit conversion was missed.

Frequently Asked Questions

What is the formula for angular acceleration? Angular Acceleration = (Final Angular Velocity − Initial Angular Velocity) ÷ Time (α = (ωf − ωi) ÷ t), typically measured in radians per second squared.

How do you calculate angular acceleration in rad/s²? Subtract the initial angular velocity from the final angular velocity (both in rad/s), then divide by the time in seconds.

What does negative angular acceleration mean? A negative angular acceleration means the object's rotation is slowing down — sometimes called angular deceleration — rather than speeding up.

How do I convert rad/s² to RPM per second? Multiply the value in rad/s² by 60 ÷ (2π) ≈ 9.5493. For example, 4 rad/s² × 9.5493 ≈ 38.2 RPM/s.

Can this calculator solve for final angular velocity or time, not just angular acceleration? Yes — switch the 'What do you want to find?' dropdown to Final Angular Velocity or Time, and the calculator automatically rearranges the formula and solves for whichever value you need.

How is angular acceleration related to tangential acceleration? Tangential (linear) acceleration equals angular acceleration multiplied by the radius: aₜ = α × r, the rotational counterpart of v = ω × r.