About the Rotational Kinematics Calculator
This rotational kinematics calculator solves constant-angular-acceleration problems — the rotational versions of the familiar SUVAT equations. Choose which three quantities you know (initial angular velocity ω₀ plus two of final angular velocity ω, angular acceleration α, time t and angular displacement θ) and it finds the other two, along with the number of revolutions turned and the final speed in rpm.
It is meant for physics and engineering students working through rotation problems, and for practical questions such as how many turns a wheel makes while speeding up, how long a fan takes to spin down, or what angular acceleration a motor needs to reach full speed in a given time. A chart traces the angular velocity and the angle turned over the course of the motion.
All quantities share one angle unit — radians, degrees or revolutions — so enter ω in that unit per second and α in that unit per second squared. The equations assume the angular acceleration is constant throughout; a negative α means the object is slowing down. If the inputs describe a motion that cannot happen (for example a slowing wheel that would need to speed up), the calculator says so rather than returning a misleading number.
How to use the rotational kinematics calculator
- 1Choose which three quantities you know.
- 2Pick the angle unit used by all your inputs.
- 3Enter the initial angular velocity and the other two known values.
- 4Use a negative angular acceleration for something that is slowing down.
- 5Read the solved values, revolutions turned and final rpm.
Formula and method
With constant angular acceleration, rotation obeys the same four equations as straight-line motion with constant acceleration, with angle θ in place of distance, angular velocity ω in place of speed and angular acceleration α in place of acceleration. Each equation leaves out one of the five quantities, so knowing any three lets you find the other two; the calculator always takes ω₀ as known and picks the equation pair that matches your choice.
Because every equation is linear in the angle unit, you can work in radians, degrees or revolutions as long as all inputs use the same one. Revolutions are θ divided by 2π (radians) or 360 (degrees), and rpm is the final angular velocity in revolutions per second times 60. Results assume α stays constant for the whole interval; real motors and brakes often deliver varying torque, so treat the answer as an average.
- θ
- Angular displacement (rad, ° or rev)
- ω₀
- Initial angular velocity (unit/s)
- ω
- Final angular velocity (unit/s)
- α
- Constant angular acceleration (unit/s²)
- t
- Elapsed time (s)
Worked examples
Wheel starting from rest at 2 rad/s² for 5 s
ω = 0 + 2 × 5 = 10 rad/s and θ = ½ × 2 × 5² = 25 rad, which is 25 / 2π = 3.98 revolutions. 10 rad/s equals 95.5 rpm.
Motor reaching 3,000 rpm in 4 s
3,000 rpm is 50 rev/s. α = (50 − 0) / 4 = 12.5 rev/s², and the motor turns θ = ½(0 + 50) × 4 = 100 revolutions while spinning up.
Fan coasting to a stop
t = (0 − 20) / (−0.5) = 40 s and θ = (0² − 20²) / (2 × −0.5) = 400 rad, or about 63.7 turns before stopping.
Speed after turning 50 rad
ω = √(5² + 2 × 2 × 50) = √225 = 15 rad/s, reached after t = (15 − 5) / 2 = 5 s.
Wheel that reverses before reaching the angle
Spinning backwards at 5 rad/s, the wheel first stops (after 2.5 s, at −6.25 rad), then speeds up forwards. Solving 50 = −5t + t² gives t = 10 s, when ω = −5 + 2 × 10 = 15 rad/s.
Turntable speeding up through one turn (degrees)
α = (270² − 90²) / (2 × 360) = 64,800 / 720 = 90 °/s², and t = 2 × 360 / (90 + 270) = 2 s.
Frequently asked questions
What are the rotational kinematics equations?+
For constant angular acceleration: ω = ω₀ + αt, θ = ω₀t + ½αt², ω² = ω₀² + 2αθ and θ = ½(ω₀ + ω)t. They mirror the linear equations of motion with θ, ω and α replacing distance, velocity and acceleration.
How do I calculate angular acceleration?+
If you know the change in angular velocity and the time, α = (ω − ω₀)/t. If you know the angle turned instead of the time, use α = (ω² − ω₀²)/(2θ).
How do I convert rpm to rad/s?+
Multiply rpm by 2π/60 ≈ 0.10472. For example, 3,000 rpm = 314.16 rad/s. To go back, multiply rad/s by 60/(2π) ≈ 9.549.
How many revolutions does a wheel make?+
Find the angular displacement θ and divide by 2π if it is in radians or by 360 if it is in degrees. The calculator reports the revolutions turned automatically for every solution.
How are angular and linear quantities related?+
For a point at radius r: arc length s = rθ, tangential speed v = rω and tangential acceleration a = rα, with θ in radians. That lets you turn wheel rotation into distance travelled.
What does a negative angular acceleration mean?+
It means the angular acceleration points opposite to the rotation, so the object is slowing down (decelerating). A fan with ω₀ = 20 rad/s and α = −0.5 rad/s² stops after 40 s.