Angular Momentum Calculator: Formula & Examples

This calculator works out the magnitude of angular momentum for a point mass moving in a circle, given its mass, velocity, and the radius of its path. You enter those three values and get the angular momentum in kilogram-meters squared per second (kg·m²/s), plus the moment of inertia and angular velocity that go with it. It uses the standard physics formula L = m·v·r, which is the most common way angular momentum is introduced and tested.

Formula

The fundamental formula for angular momentum of a point mass moving in a circle is:

L = m × v × r

where:

• L = angular momentum (kg·m²/s)
• m = mass of the object (kg)
• v = linear (tangential) velocity (m/s)
• r = radius of the circular path (m)

This is the simplified version of the general definition L = I × ω, where I is the moment of inertia and ω is the angular velocity. For a point mass, I = m·r² and ω = v/r, so substituting gives L = (m·r²)·(v/r) = m·v·r. The calculator uses the direct form because it is the easiest to measure in many introductory problems.

Worked Example

Using the default values from the calculator (mass = 2 kg, velocity = 5 m/s, radius = 1.5 m):

1. Write the formula: L = m × v × r
2. Substitute the numbers: L = 2 × 5 × 1.5
3. Multiply: L = 10 × 1.5 = 15
4. Add units: L = 15 kg·m²/s

The calculator returns 15.00 kg·m²/s, which matches. The moment of inertia is I = 2 × (1.5)² = 4.5 kg·m², and the angular velocity is ω = 5 / 1.5 ≈ 3.33 rad/s. You can verify that L = I × ω = 4.5 × 3.33 ≈ 15.

When to Use This Formula

This formula applies when you have a single object (or a small object compared to the radius) moving in a circle at a constant speed. Common situations include:

• A ball on a string being swung overhead
• A satellite in a circular orbit (approximate)
• A car going around a circular track
• An electron in a simple Bohr-model atom

For extended objects like spinning disks, planets, or figure skaters, you need the moment of inertia of the whole shape, not just m·r². For example, a solid disk has I = (1/2)·M·R², so L = (1/2)·M·R²·ω. The calculator here is for the point-mass case, which is the most common homework and conceptual problem.

How to Read the Result

Angular momentum is a vector quantity, but this calculator gives its magnitude. The direction is along the axis of rotation, given by the right-hand rule: curl your fingers in the direction of motion and your thumb points along the angular momentum vector. The unit kg·m²/s is equivalent to J·s (joule-seconds), which you might see in quantum mechanics.

A larger angular momentum means the object is harder to stop spinning. Doubling the mass, velocity, or radius each doubles L. Because the formula is linear in all three variables, the effect is easy to predict.

Common Mistakes and Misconceptions

• Using diameter instead of radius: The radius is half the diameter. If a problem gives a diameter of 3 m, the radius is 1.5 m. Using 3 m gives double the correct answer.
• Mixing units: Always convert to kilograms, meters, and seconds. If mass is given in grams, divide by 1000. If velocity is in km/h, divide by 3.6 to get m/s.
• Confusing angular momentum with moment of inertia: They are different. Moment of inertia (kg·m²) measures how mass is distributed; angular momentum (kg·m²/s) measures the actual rotational motion.
• Using linear momentum formula: Linear momentum is p = m·v, with units kg·m/s. Angular momentum multiplies by the radius, so it has an extra meter.
• Forgetting that velocity is tangential: The velocity in the formula is the speed along the circle, not the angular velocity (rad/s). If you are given angular velocity, convert to linear velocity with v = ω·r.

Conservation of Angular Momentum

The reason angular momentum is so important is that it is conserved in a closed system (no external torque). This explains many everyday phenomena:

• A figure skater pulls in their arms to spin faster: reducing r reduces I, so ω increases to keep L constant.
• A diver tucks to somersault faster in the air.
• A spinning ice skater extends arms to slow down.
• Planets orbit faster when closer to the Sun (Kepler’s second law).

In all these cases, L stays the same if no external torque acts. The calculator helps you see the trade-off: if you halve the radius, the velocity must double to keep L constant (assuming mass stays the same).

Limits of the Formula

The formula L = m·v·r assumes the object is a point mass and moves in a perfect circle at constant speed. Real objects have size, so their moment of inertia is not exactly m·r². For a sphere rolling without slipping, you need to add the rotational contribution. For elliptical orbits, the radius changes, and you need the full vector cross product L = r × p. At relativistic speeds (close to the speed of light), the formula also breaks down. For most introductory physics problems, however, this simple version is exactly what is expected.

Frequently Asked Questions

What is angular momentum in simple words?
Angular momentum is a measure of how much rotational motion an object has, similar to how linear momentum measures straight-line motion. It depends on how much mass is moving, how fast it moves, and how far it is from the axis of rotation.

How do I find angular momentum from angular velocity?
Use L = I·ω, where I is the moment of inertia and ω is the angular velocity in rad/s. For a point mass, I = m·r², so you can also use L = m·r²·ω. If you have linear velocity, use L = m·v·r.

What is the unit of angular momentum?
The SI unit is kilogram-meter squared per second (kg·m²/s). In quantum mechanics, it is often expressed in units of ħ (h-bar), which is Planck’s constant divided by 2Ï€.

Is angular momentum always conserved?
Angular momentum is conserved when the net external torque on a system is zero. That is true for isolated systems, like a spinning skater or a planet orbiting the Sun (ignoring tidal effects). If an external torque acts, like friction on a spinning wheel, angular momentum changes.

What is the difference between angular momentum and moment of inertia?
Moment of inertia (I) is a property of the object’s mass distribution, with units kg·m². Angular momentum (L) is the actual rotational motion, with units kg·m²/s. They are related by L = I·ω, so the same object can have different angular momentum depending on its angular velocity.

Can angular momentum be negative?
Yes, if you define a coordinate system. The magnitude is always positive, but the vector can point in the opposite direction. In one-dimensional problems, clockwise rotation is often taken as negative and counterclockwise as positive.

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in m | Mass | kg | 2 | 0
in v | Velocity | m/s | 5 | 0
in r | Radius | m | 1.5 | 0
out L | Angular momentum | kg·m²/s | m * v * r | 2
out I | Moment of inertia | kg·m² | m * r^2 | 2
out omega | Angular velocity | rad/s | v / r | 2