Rotational kinematics equations for constant angular acceleration
Rotational kinematics describes fixed-axis angular motion without calculating the torques that cause it.
The familiar linear kinematic equations have rotational counterparts using angle, angular velocity, and angular acceleration.
How to use the rotational kinematics calculator
- Select the unknown: Choose displacement, velocity, acceleration, or time.
- Enter the known values: Use radians, seconds, and one consistent positive rotation direction.
- Calculate: Review the result, unit, and equation used.
Formula and variables
These relationships apply when angular acceleration stays constant throughout the elapsed time.
ωf = ω₀ + αt; Δθ = ω₀t + ½αt²- ω₀ — Initial angular velocity
- Starting rotation rate (rad/s)
- ωf — Final angular velocity
- Ending rotation rate (rad/s)
- α — Angular acceleration
- Constant change in angular velocity (rad/s²)
- Δθ — Angular displacement
- Change in angular position (rad)
- t — Time
- Elapsed time (s)
Accelerating rotor
A rotor starts from rest and accelerates at 2 rad/s² for 10 seconds.
- ω₀
- 0 rad/s
- α
- 2 rad/s²
- t
- 10 s
- ωf = ω₀ + αt = 0 + 2(10)
- Δθ = ω₀t + ½αt² = 100
Result: Final angular velocity is 20 rad/s and displacement is 100 rad.
Both results rely on acceleration remaining constant for the full interval.
Understanding your results
Signs describe rotational direction
Choose clockwise or counterclockwise as positive before entering values, then keep that convention for every input.
- A negative angular velocity points opposite the chosen positive direction.
- Acceleration opposite velocity slows the rotation until the sign changes.
Assumptions
- Fixed-axis rotation.
- Constant angular acceleration.
- Elapsed time begins at zero.
- Angles are expressed in radians.
Limitations
- Does not model time-varying angular acceleration.
- Does not calculate torque, moment of inertia, friction, or energy.
- Does not enumerate multiple mathematical branches from squared-velocity equations.
Common mistakes
- Entering degrees where radians are required.
- Mixing revolutions per minute with radians per second.
- Using inconsistent signs for angular quantities.
- Applying constant-acceleration equations to variable acceleration.
Practical use cases
Physics coursework
Solve wheel, disk, flywheel, and rotor kinematics problems.
Preliminary engineering checks
Estimate idealized angular motion before a dynamics analysis.
Frequently asked questions
Are rotation equations and rotational kinematics the same topic?
Usually yes. Rotation equations commonly refers to the constant-angular-acceleration kinematic equations collected here.
Why must angles use radians?
Radians make relationships such as arc length s = rθ and tangential speed v = rω directly coherent without an additional conversion factor.
Sources and review
- Rotation with Constant Angular Acceleration — OpenStax University Physics Volume 1. Accessed 2026-07-14.
Reviewed 2026-07-14.