Rotational Motion & Angular Momentum Calculator
Solve rotation problems. Give any three of θ, ω₀, ω, α and t for the other two, work out angular momentum L = Iω and rotational energy ½Iω², conserve angular momentum when a spinning body changes shape, and roll a ring, disc or sphere down a slope.
How to Use
- Pick a mode: rotational kinematics, angular momentum and energy, conservation of angular momentum, or rolling down a slope.
- In Kinematics, choose the two unknowns from the list and enter the other three of angle θ, starting and final angular velocity ω₀ and ω, angular acceleration α and time t. Slowing down is a negative α.
- Pick a unit beside each value: degrees, radians or revolutions for angles; rad/s, rpm or rev/s for spin; kg·m² or lb·ft² for moment of inertia.
- For Rolling, choose the shape (ring, disc, solid or hollow sphere), the slope angle and length, and the radius.
- Read the answers in the highlighted fields and readouts, and the drawing: a wheel and its ω–t graph, a spinning disc, a skater, or a race of every shape down your slope.
- Press a preset to load an example and check Show Work for each step.
Worked Example
A fan spinning up. From rest to 1,200 rpm in 4 s. Convert first: 1,200 rpm = 1,200 × 2π ÷ 60 = 125.66 rad/s. Then α = (ω − ω₀) ÷ t = 125.66 ÷ 4 = 31.42 rad/s² and θ = ½(ω₀ + ω)t = ½ × 125.66 × 4 = 251.3 rad, which is 40 revolutions.
A skater pulling her arms in. With I falling from 4 to 1.6 kg·m² (illustrative) at 1 rev/s, I₁ω₁ = I₂ω₂ gives ω₂ = 4 × 1 ÷ 1.6 = 2.5 rev/s. Angular momentum is conserved but energy is not: ½Iω² rises from 78.96 J to 197.4 J, the 118.4 J coming from her muscles.
The common mistake: forgetting the spin energy when rolling. A solid sphere rolling down a 2 m slope at 30° drops 1 m. Setting mgh = ½mv² gives v = √(2 × 9.80665 × 1) = 4.429 m/s, the speed of something sliding without friction. But part of the energy goes into spinning, so v = √(2gh ÷ (1 + 2/5)) = 3.743 m/s; a ring, with half its energy in spin, only reaches 3.132 m/s.
Show Work
Formulas
From Kepler’s Areas to Euler’s Rigid Bodies
Conservation of angular momentum appeared first in the sky. Johannes Kepler’s second law (1609) says a planet sweeps out equal areas in equal times, speeding up near the Sun and slowing far from it, which is a planet keeping its angular momentum constant. Isaac Newton showed in the Principia (1687) that this follows for any force pointing at a fixed centre.
Galileo timed balls rolling down inclined planes for Two New Sciences (1638). He did not know that a rolling ball only accelerates at 5/7 of the rate of a frictionless slider, but because that factor is the same all the way down, his distances still grew with the square of the time.
Leonhard Euler gave rotation its full mechanics in his 1765 book on the motion of rigid bodies, where he named the moment of inertia and wrote the equations that link torque, I and angular acceleration. The same equations explain why a skater, a diver tucking into a somersault and a collapsing star all spin up as they draw their mass inwards.
About This Tool
This calculator puts four kinds of rotation problem on one page. Kinematics solves any two of θ, ω₀, ω, α and t from the other three, including the two-root cases where a spinning body reverses and passes the same angle twice. L = Iω gives angular momentum, moment of inertia or angular velocity with the rotational energy. Conservation follows a body that changes its moment of inertia, and Rolling takes a ring, disc or sphere down a slope, with the grip it needs and a race against every other shape. Angular velocity can be entered in rad/s, rpm, rev/s or degrees per second; the conversions are in the working.
Everything runs in your browser; nothing you enter is sent anywhere.
Related tools: Moment of Inertia & Centroid Calculator, Torque Calculator, and Circular Motion Calculator.
Frequently Asked Questions
What are the equations of rotational motion?
They copy the straight-line ones with θ for distance, ω for velocity and α for acceleration: ω = ω₀ + αt, θ = ω₀t + ½αt², ω² = ω₀² + 2αθ and θ = ½(ω₀ + ω)t. A fan going from rest to 1,200 rpm (125.66 rad/s) in 4 s has α = 31.42 rad/s² and turns θ = ½ × 125.66 × 4 = 251.3 rad, exactly 40 revolutions.
How do I calculate angular momentum and rotational kinetic energy?
L = Iω and KE = ½Iω², with ω in rad/s. A 0.5 kg·m² flywheel at 3,000 rpm (314.16 rad/s) has L = 157.1 kg·m²/s and KE = 24.67 kJ. Doubling the speed doubles L but quadruples the energy, to 98.70 kJ.
Why does a skater spin faster when she pulls her arms in?
With no outside torque, L = Iω stays the same, so a smaller I means a larger ω. Going from I = 4 to 1.6 kg·m² (illustrative figures) at 1 rev/s gives 4 × 1 ÷ 1.6 = 2.5 rev/s. Her spin energy rises from 78.96 J to 197.4 J; the extra 118.4 J is the work her arms do pulling inwards.
Which shape rolls down a slope fastest?
The one with the smallest I ÷ mr², whatever its mass or size: a = g sin θ ÷ (1 + I/mr²). Down a 2 m slope at 30°, a solid sphere takes 1.069 s, a solid disc 1.106 s, a hollow sphere 1.166 s and a ring 1.277 s; a frictionless sliding block would take 0.903 s.
What does rolling without slipping mean?
The contact point does not skid, so the centre moves at v = ωr; the bottom of the wheel is momentarily at rest and the top moves at 2v. A bicycle wheel of 0.34 m radius at 25 km/h (6.944 m/s) spins at ω = 6.944 ÷ 0.34 = 20.42 rad/s, which is 195.0 rpm.
How do I use the Rotational Motion & Angular Momentum Calculator?
Just type your numbers. The answer shows up right away — there is no button to press. Change anything and it updates by itself.
Do I need to install or sign up for anything?
Not at all — it runs in the browser with nothing to install and no account. After it loads once, it even works without an internet connection.
Is my information private?
Yes. Everything happens in your browser. Nothing you type is sent to a server or saved anywhere.
Common Use Cases
Motors and fans
A fan reaching 1,200 rpm in 4 s from rest accelerates at 31.42 rad/s² and makes 40 turns while spinning up, averaging 600 rpm.
Braking a wheel
A wheel at 30 rad/s braking at 2 rad/s² stops in 15 s after 225 rad, 35.81 turns.
Flywheel energy storage
A 0.5 kg·m² flywheel at 3,000 rpm stores 24.67 kJ and carries 157.1 kg·m²/s of angular momentum.
Figure skating and diving
Cutting the moment of inertia from 4 to 1.6 kg·m² (illustrative) turns 1 rev/s into 2.5 rev/s, which is how a skater speeds up a spin or a diver a somersault.
Rolling races
A can (solid cylinder) rolling 1.5 m down a 10° ramp takes 1.626 s and reaches 1.845 m/s; a 3.3 cm radius means it spins at 534 rpm at the bottom.
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