🌀 Angular Momentum Calculator
L = I × ω — Calculate angular momentum, inertia, or angular velocity.
Result
Conservation of Angular Momentum
Find ω₂ when moment of inertia changes (I₁ω₁ = I₂ω₂)
ω₂ (New Angular Velocity)
How to Use This Calculator
The top section solves L = Iω for any of the three variables. The bottom section handles conservation of angular momentum: enter the initial I₁ and ω₁ plus a new moment of inertia I₂, and it tells you the new angular velocity ω₂. All angular velocities can be entered in rad/s; the results also show the equivalent in RPM.
Select whether you want to find L, I, or ω from the dropdown.
Enter moment of inertia I in kg·m². For a solid disk of mass 2 kg and radius 0.2 m, I = ½ × 2 × 0.04 = 0.04 kg·m².
Enter angular velocity ω in rad/s. To convert from RPM: ω = RPM × 2π / 60.
For conservation problems, scroll down and enter I₁, ω₁, and the new I₂. The calculator finds the resulting ω₂.
Angular Momentum Formula
L is angular momentum, I is moment of inertia, and ω is angular velocity. The conservation law says: if no external torque acts on a system, the product I × ω stays the same. Reduce I and ω must increase to keep L constant. Increase I and ω must decrease.
Worked Examples
Where This Comes Up in Real Life
Divers and gymnasts use angular momentum conservation to control their spin. A diver leaving the board with arms spread starts rotating slowly. Tucking the body reduces I dramatically, which spins them up fast enough to complete multiple somersaults before entry. Straightening out again increases I and slows the spin so they can enter the water cleanly.
Gyroscopes in aeroplanes and spacecraft maintain orientation because any external torque creates a change in the direction of L, not just its magnitude. The spinning rotor resists any attempt to change its orientation, which is exactly what makes a gyroscope useful for navigation. In astrophysics, neutron stars that collapse from a large stellar radius to about 10 km can spin at hundreds of revolutions per second because I shrinks by a factor of billions while L is nearly conserved.
Frequently Asked Questions
What is angular momentum?
Angular momentum L is the rotational equivalent of linear momentum. L = I × ω, where I is moment of inertia (kg·m²) and ω is angular velocity (rad/s).
How is angular momentum conserved?
If no external torque acts on a system, angular momentum is conserved: I₁ω₁ = I₂ω₂. This explains why a spinning ice skater speeds up when pulling in their arms.
What is the relationship between L and linear momentum?
For a point mass: L = m × v × r × sin(θ), where r is the distance from the axis and θ is the angle between v and r.
What is the unit of angular momentum?
kg·m²/s in SI.
What is angular impulse?
Angular impulse = τ × Δt = ΔL. Just as force × time changes linear momentum, torque × time changes angular momentum.