Angular Momentum Calculator
Angular momentum of a spinning body or a moving point mass, and the new spin rate when the moment of inertia changes.
⛸️ What is Angular Momentum?
Angular momentum is the rotational counterpart of linear momentum. It measures how much rotation a body or a moving particle carries, and it depends on both how fast something turns and how its mass is arranged around the axis. For a rigid body spinning about a fixed axis, L = I ω, the product of the moment of inertia I and the angular velocity ω. For a single particle moving about a point, L = r m v sinθ, which is the particle's linear momentum m v multiplied by the perpendicular distance from the point to the line of motion. The SI unit is kg m2/s.
The reason angular momentum matters is that it is conserved. If no net external torque acts on a system, its total angular momentum cannot change. This explains why a figure skater spins faster when the arms are pulled in, why a diver tucks to complete a somersault, why planets move faster at the closest point of their orbit. It also underlies the stability of gyroscopes, the design of flywheels and reaction wheels, and the way a collapsing star spins up. The flywheel energy storage calculator uses the same moment of inertia ideas.
This calculator has three modes. The spinning body mode finds L from a moment of inertia, either typed in or computed from a disk, hoop, sphere or rod, and an angular speed in rad/s, rpm or degrees per second. The point mass mode applies L = r m v sinθ, showing the lever arm and the maximum L that would apply if the velocity were perpendicular. The conservation mode solves I1ω1 = I2ω2 for the final spin and also reports the kinetic energy before and after.
A common misconception is that conservation of angular momentum means conservation of spin. The spin changes whenever I changes. Another is that energy is conserved in the skater example. It is not: pulling the arms in requires work, and kinetic energy rises while L stays fixed. Angular momentum also depends on the chosen origin, so always state the point or axis. The related angular velocity calculator and angular acceleration calculator cover the other rotational quantities, and the momentum calculator handles straight-line motion.
📐 Formula
📖 How to Use This Calculator
Steps
💡 Example Calculations
Example 1 — A Spinning Flywheel Disk
A solid disk of mass 20 kg and radius 0.3 m spins at 1500 rpm
Example 2 — A Hoop
A thin hoop of mass 2 kg and radius 0.3 m turns at 120 rpm
Example 3 — A Ball on a String
A 0.5 kg ball moves at 4 m/s in a circle of radius 1.2 m, so the velocity is perpendicular to the radius
Example 4 — A Satellite in a Circular Orbit
A 500 kg satellite moves at 7500 m/s at a distance of 6 900 000 m from the centre of its planet, perpendicular to the radius
Example 5 — Figure Skater Pulling In the Arms
A skater spins at 60 rpm with I = 3.0 kg m² (illustrative values), then pulls in the arms so I becomes 0.9 kg m²
Example 6 — A Diver Tucking
A diver leaves the board at 30 rpm with I = 14 kg m² (illustrative) and tucks to I = 3.5 kg m²
❓ Frequently Asked Questions
🔗 Related Calculators
What is the formula for angular momentum?
For a rigid body spinning about a fixed axis, L = I omega, the moment of inertia times the angular velocity, in kg m squared per second. For a point mass, L = r m v sin(theta), where theta is the angle between the position vector and the velocity.
What are the units of angular momentum?
The SI unit is kilogram metre squared per second (kg m squared per s), which is the same as joule second. It is the product of moment of inertia (kg m squared) and angular velocity (rad/s).
How does a figure skater spin faster?
Pulling the arms in reduces the moment of inertia. With almost no external torque, angular momentum I omega stays constant, so omega must increase. An example: I falls from 3.0 to 0.9 kg m squared and the spin rises from 60 rpm to 200 rpm.
What is conservation of angular momentum?
When the net external torque on a system is zero, its total angular momentum does not change. Internal forces can move mass closer to or farther from the axis, but they change I and omega in opposite ways so that I omega stays the same.
Does the skater gain energy when pulling in the arms?
Yes. Rotational kinetic energy is L squared over 2 I, so reducing I raises the energy. In the example, kinetic energy goes from 59.2 J to 197.4 J. The extra 138.2 J is work done by the skater's muscles against the outward pull of the arms.
How do I calculate angular momentum from rpm?
Convert rpm to rad/s by multiplying by 2 pi / 60, then multiply by the moment of inertia. A 0.9 kg m squared disk at 1500 rpm (157.08 rad/s) has L = 0.9 x 157.08 = 141.37 kg m squared per s.
What is the moment of inertia of common shapes?
A solid disk is one half M R squared, a hoop is M R squared, a solid sphere is two fifths M R squared, a rod about its centre is M L squared over 12 and a rod about one end is M L squared over 3.
Can a straight-line moving object have angular momentum?
Yes, relative to a chosen point. A particle moving in a straight line has L = r m v sin(theta) about any point not on its line of motion. L depends on the point chosen as the origin.
When is angular momentum largest for a point mass?
When the velocity is perpendicular to the position vector, theta = 90 degrees, as in circular motion. Then L = r m v. For a 0.5 kg ball at 4 m/s on a 1.2 m string, L = 2.4 kg m squared per s.
Is angular momentum a vector?
Yes. It points along the rotation axis by the right-hand rule. This calculator reports the magnitude and uses a sign to show the direction of rotation about a single axis.
How is angular momentum different from linear momentum?
Linear momentum p = m v depends on mass and velocity. Angular momentum also depends on where the mass is relative to the axis. The same velocity farther from the axis carries more angular momentum.