Angular Momentum Calculator

TL;DR: To calculate angular momentum, multiply the object’s mass (m), tangential velocity (v), and radius of rotation (r) using the formula L = m × v × r, so for a 2 kg object moving at 5 m/s on a 1 m radius path, the angular momentum is 10 kg·m²/s.

What Is the Angular Momentum Calculator?

The Angular Momentum Calculator is a free online tool that computes the rotational equivalent of linear momentum. While linear momentum (p = m × v) describes how much “oomph” an object has moving in a straight line, angular momentum (L) quantifies how much rotational “oomph” an object has as it spins or orbits. The calculator requires three physical inputs—mass (in kilograms), tangential velocity (in meters per second), and radius (in meters)—and instantly produces the angular momentum in kilogram-meters squared per second (kg·m²/s).

This tool is essential for physics students tackling rotational dynamics problems, engineers designing flywheels or turbines, and even astronomers estimating the spin of planets or binary star systems. For example, a figure skater pulling in their arms to spin faster is a classic demonstration of angular momentum conservation—the skater reduces their radius (r), which increases their rotation speed (v) while L stays constant. The calculator lets you model such changes numerically instead of guessing.

The primary output is the scalar magnitude of angular momentum for an object moving in a circular path. This assumes the velocity is purely tangential (perpendicular to the radius). The calculator is designed for simplicity and speed—enter your three values, click calculate, and you get the result immediately without any subscription or software download.

How to Use the Calculator

Using the Angular Momentum Calculator takes less than ten seconds. Follow these five steps:

1. Enter mass (m): Type the object's mass into the first field, using kilograms (kg). For instance, type 2 for a 2 kg object.
2. Enter tangential velocity (v): Input the speed perpendicular to the radius in meters per second (m/s). Example: type 5.
3. Enter radius (r): Specify the distance from the rotation axis to the object in meters (m). Example: type 1.
4. Click “Calculate”: Press the calculate button. The tool instantly performs the multiplication L = m × v × r.
5. Read the result: The output field displays the angular momentum in kg·m²/s. For the example above, the result is 10 kg·m²/s.

If you change any input value, simply re-click calculate. The calculator supports decimal values (e.g., 2.5 kg, 0.75 m/s, 3.14 m) for higher precision. There is no limit on the number of calculations you can perform.

Formula and Calculation Method

The calculator uses the fundamental angular momentum formula for a point mass moving in a circular path:

L = m × v × r

Where:

• L = angular momentum (kg·m²/s)
• m = mass of the object (kg)
• v = tangential velocity (m/s)—speed along the circle's edge, perpendicular to the radius
• r = radius (m)—distance from the center of rotation to the object

This formula is derived from the vector definition L = r × p, where p is linear momentum (m·v). Because the radius and velocity vectors are perpendicular for circular motion, the cross product simplifies to simple multiplication: L = r × (m × v).

Worked example: Suppose you have a 2 kg ball attached to a 1-meter rope, whirled overhead at a tangential speed of 5 m/s.

Step 1: Identify the inputs—m = 2 kg, v = 5 m/s, r = 1 m.
Step 2: Apply the formula—L = 2 × 5 × 1.
Step 3: Multiply—L = 10 kg·m²/s.

The magnitude 10 kg·m²/s means the ball possesses 10 units of rotational momentum. If you shorten the rope to 0.5 m while keeping the same tangential speed, L drops to 5 kg·m²/s—unless the speed increases to compensate, which is exactly how angular momentum conservation works in real systems.

Practical Examples

Here are three realistic scenarios demonstrating how different mass, velocity, and radius combinations affect the output. Each uses the same formula but produces physically distinct results.

Figure skater: A 50 kg skater spinning with arms extended has a radius of 0.5 m and tangential arm speed of 3 m/s. L = 50 × 3 × 0.5 = 75 kg·m²/s. When they pull their arms in, their radius halves to 0.25 m—to conserve L (75), their tangential speed doubles to 6 m/s, which is why they spin faster.

Car on a roundabout: A 1200 kg car moving at 8 m/s on a circular track with a 10 m radius has L = 1200 × 8 × 10 = 96,000 kg·m²/s. This massive value shows why heavy vehicles require substantial steering forces to change their rotational state.

Electron model: In Bohr's simple model, an electron (9.11 × 10⁻³¹ kg) orbits a nucleus at 2.2 × 10⁶ m/s with a radius of 5.3 × 10⁻¹¹ m. The product yields approximately 1.06 × 10⁻³⁴ kg·m²/s—remarkably close to Planck's constant divided by 2π (1.055 × 10⁻³⁴), a famous quantum mechanical coincidence.

Tips for Accurate Results

To get meaningful results from the calculator, pay attention to the following factors based on the actual input fields:

• Use consistent SI units: The calculator assumes kilograms, meters per second, and meters. If your data is in grams, convert to kilograms (divide by 1000). If velocity is in km/h, convert to m/s (divide by 3.6). Mixing units will produce nonsensical output.
• Check realistic ranges: Verify your inputs reflect physical reality. A mass of 0 kg or a radius of 0 m makes no sense. Similarly, a tangential velocity of 10,000 m/s for a bicycle wheel suggests a unit error—realistic bicycle speeds are 5–15 m/s.
• Avoid zero or negative values: The formula requires positive numbers for mass, velocity, and radius. Any zero yields an L of zero (meaning no rotation), and negative values would imply impossible negative mass or direction. Double-check your entries if you see unexpected results.
• Don't round intermediate steps: If you're hand-verifying the calculator's output, do not round m, v, or r before multiplying. For example, using v = 9.81 m/s and rounding to 9.8 before multiplying introduces a 0.1% error that grows with large values.
• Confirm tangential velocity, not angular velocity: This calculator expects linear tangential speed (m/s), not angular velocity (rad/s). If you have angular velocity ω, convert first using v = ω × r before entering values.

Frequently Asked Questions

What is the difference between angular momentum and linear momentum?

Linear momentum (p = m × v) describes motion along a straight path—like a car driving down a road. It depends only on mass and velocity. Angular momentum (L = m × v × r) describes rotational motion and adds a third factor: the distance from the rotation axis (radius). The key difference is that angular momentum is conserved when no external torque acts, even if the shape changes. For example, a spinning ice skater has constant linear momentum (if moving slowly across the ice) but changing angular momentum distribution—when they pull their arms in, radius decreases, velocity increases, but L stays the same. In SI units, linear momentum is kg·m/s while angular momentum is kg·m²/s.

Can I calculate angular momentum if I only know angular velocity but not tangential velocity?

Yes, but you must convert angular velocity to tangential velocity first. Angular velocity (ω) is measured in radians per second. The relationship is v = ω × r, where r is the radius. Substitute this into the calculator's formula: L = m × (ω × r) × r = m × ω × r². If you have a 3 kg object with an angular velocity of 2 rad/s at a radius of 4 m, first calculate v = 2 × 4 = 8 m/s, then enter m = 3, v = 8, r = 4 into the calculator. Alternatively, use the expanded formula directly: L = 3 × 2 × 16 = 96 kg·m²/s.

Why do my results look extremely large or tiny, and is that normal?

Angular momentum values span an enormous range depending on the system. A typical dust particle (10⁻⁹ kg) orbiting at 1 m/s with a 0.01 m radius gives L = 10⁻¹¹ kg·m²/s—a tiny number. A planet like Earth (6 × 10²⁴ kg) has L ≈ 7 × 10³³ kg·m²/s—an enormous number. If your result seems wrong, check your units first: grams instead of kilograms inflate results by 1000×, while kilometers instead of meters deflate them by 1000×. Also verify the radius is measured from the actual rotation axis, not from the object's center. For a spinning top, the radius is the distance from the top's point to its center of mass, not the top's full height. Using scientific notation for very large or small numbers (e.g., 9.11e-31) is perfectly valid—just ensure the calculator you're using supports exponent notation.