What this calculator does
Find out how much spinning motion an object carries, called its angular momentum. It's the spinning version of ordinary momentum. Ordinary momentum is mass times velocity. Angular momentum is the moment of inertia times the spin rate (angular velocity), L = I × ω. Here L is angular momentum, I is the moment of inertia and ω (omega) is the spin rate. The moment of inertia does the job that mass does for straight-line motion. It measures how hard an object is to spin up or slow down. It depends on how much mass there is, and also on how far that mass sits from the axis it turns around.
This calculator works three ways. If you already know the moment of inertia, enter it with the spin rate. If you have a common solid shape, such as a ball, a disc, a ring or a rod, enter its mass and size. The calculator then uses the standard moment-of-inertia formula for that shape. For a small object moving around a center, like a ball on a string, it uses L = m × v × r (mass × speed × distance). Every result also shows the moment of inertia and the spin rate in rpm. It also shows the rotational kinetic energy, the energy stored in the spin.
Where it's used
Why this math matters outside the classroom:
- Figure skating spins. A skater who pulls in her arms halves her moment of inertia, so to keep angular momentum constant her spin rate doubles.
- Gyroscopes. Navigation gyroscopes keep a fast-spinning rotor's angular momentum pointing the same way, giving aircraft and spacecraft a fixed reference.
- Spacecraft pointing. Spacecraft turn by speeding up an internal reaction wheel, which spins the craft the opposite way because total angular momentum stays the same.
- Ball on a string. A 2 kg ball on a 0.5 m string moving at 3 m/s has angular momentum mvr = 2 × 3 × 0.5 = 3 kg·m²/s.
- Diving somersaults. Divers tuck tightly to shrink their moment of inertia, which lets them complete more somersaults before reaching the water.
How to use it
- Choose how to describe the object: a known moment of inertia, a standard shape, or a single small mass (a point mass) moving on a circle.
- For a shape, pick it from the list. Then enter its mass and radius (or length, for a rod).
- Enter the spin rate (angular velocity). For a point mass, enter its speed and its distance from the axis instead.
- Pick the unit you want, then read the angular momentum and the values that go with it.
What the inputs mean
- Shape
- Each formula assumes the material is spread evenly (uniform density). The sphere, cylinder and hoop spin about an axis through their center. Rods turn about an axis at right angles to their length.
- Moment of inertia (I)
- The moment of inertia about the axis the object turns around. 1 lb·ft² is about 0.04214 kg·m².
- Speed at right angles to the radius (v)
- Only the part of the motion at right angles to the line from the axis counts. For motion around a circle, that's the whole speed.
- Angular velocity (ω)
- How fast it spins: in radians per second, degrees per second, rpm or revolutions per second. One full turn is 2π radians, about 6.28.
Formula and method
L = I × ω L = m × v × r (point mass, v at right angles to r) I = k × m × r², k = 2/5, 2/3, 1/2 or 1 I = m × L² ÷ 12 (rod about center), m × L² ÷ 3 (rod about end) E = ½ × I × ω² = ½ × L × ω
L is angular momentum, I is the moment of inertia and ω is the spin rate in radians per second. m is mass, v is speed and r is the distance from the axis (the radius). In the rod formulas, L stands for the rod's length. k is a shape factor, and E is the rotational kinetic energy.
The calculator first converts every input to metric (SI) units. So angular momentum comes out in kg·m²/s, the same unit as the joule second. A point mass at distance r has I = m × r² and ω = v ÷ r. So the two main formulas agree: I × ω = m × r² × v ÷ r = m × v × r.
The shape factors come from adding up m × r² over every bit of the object. A hoop has all its mass at the rim, so k = 1. A solid disc has much of its mass near the middle, so k = 1/2. A solid ball has even more mass close to the center, so k = 2/5.
Worked example
A solid disc of mass 2 kg and radius 0.5 m spins at 300 rpm. What is its angular momentum?
- Moment of inertia of a disc: I = 1/2 × 2 × 0.5² = 0.25 kg·m².
- Convert the spin rate: 300 rpm × 0.10471975512 = 31.415927 rad/s.
- Angular momentum: L = 0.25 × 31.415927 = 7.853982 kg·m²/s.
- Its rotational kinetic energy is 0.5 × 7.853982 × 31.415927 = 123.370055 J.
Result: The disc carries about 7.854 kg·m²/s of angular momentum (exactly 2.5π) and about 123.37 J of rotational energy.
Understanding the result
With no outside twisting force (torque), angular momentum stays the same. That's why a skater spins faster when she pulls her arms in. Her moment of inertia drops, so ω must rise to keep L the same.
Two objects with the same mass and spin rate can carry very different angular momentum. Moving mass outward, as in a flywheel with a heavy rim, raises I. It then stores more angular momentum and energy at the same rpm.
Assumptions and limitations
- The object turns about a single fixed axis, and the value shown is the part along that axis.
- Standard shapes have evenly spread mass and ideal forms (thin hoops, thin rods, perfect spheres).
- Speeds are far below the speed of light.
- Objects that spin about more than one axis at once, or wobble, need a fuller description called the inertia tensor.
- For a point mass, the speed must be at right angles to the radius. For any other angle, multiply by the sine of that angle first.
- For shapes not in the list, work out the moment of inertia separately, then enter it directly.
Frequently asked questions
What is the unit of angular momentum?
In metric (SI) units it's the kilogram square meter per second (kg·m²/s). That's the same as the joule second (J·s). In US units it can be written as pound square feet per second.
How is angular momentum different from linear momentum?
Linear momentum, m × v, describes motion along a line, and a force changes it. Angular momentum, I × ω, describes turning about an axis, and a torque (twisting force) changes it. Both stay the same when no outside push or twist acts.
Why does a solid ball have a smaller moment of inertia than a hoop of the same mass and radius?
The moment of inertia counts each bit of mass by the square of its distance from the axis. A hoop keeps all its mass at the full radius. Much of a ball's mass lies close to the center. So the ball's total is only 2/5 of the hoop's.
Does a point mass moving in a straight line have angular momentum?
Yes, about any point that isn't on its path. The amount is m × v × d, where d is the shortest distance from the point to the path. It stays the same as the object moves along.
How this calculator is checked
Method: Rigid-body rotation about a fixed axis (L = Iω, E = ½Iω²), point-mass angular momentum L = mvr, and standard moments of inertia for uniform solids. The calculation is covered by automated tests with known reference values and edge cases. Last updated 2026-10-18. Found a problem? Tell us.
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