Angular Momentum Calculator & Formula

This angular momentum calculator works out the angular momentum (L) of a rotating object from its moment of inertia (I) and angular velocity (ω). You enter those two values in the units of your choice, and it returns the angular momentum in kilogram meters squared per second (kg·m²/s), along with the equivalent in other common units.

Formula

The fundamental equation for angular momentum of a rigid body rotating about a fixed axis is:

L = I × ω

• L = angular momentum (kg·m²/s)
• I = moment of inertia (kg·m²) – a measure of how mass is distributed relative to the rotation axis
• ω = angular velocity (rad/s) – how fast it rotates

This is the rotational analog of linear momentum p = mv. The calculator uses exactly this formula. If you have angular velocity in revolutions per minute (rpm) instead of rad/s, convert first: ω (rad/s) = rpm × 2π / 60. The second output shows the same angular momentum expressed in kg·m²·rpm, which is useful when you input rpm directly.

Worked Example

Using the default values from the calculator: I = 2.5 kg·m², ω = 6.28 rad/s (which is almost exactly 1 revolution per second, since 2π ≈ 6.283).

1. Write down the formula: L = I × ω
2. Plug in the numbers: L = 2.5 × 6.28
3. Multiply: L = 15.7 kg·m²/s

The calculator shows 15.70 kg·m²/s, matching the hand calculation. The second output, 15.70 × 60 / (2π) = 150.0 kg·m²·rpm, is the same angular momentum expressed in different units – it’s not a different physical quantity.

When to Use This Calculator

You would use this in physics homework, engineering design, or any situation where you need the rotational equivalent of momentum. Common scenarios include:

• Calculating the angular momentum of a flywheel, turbine, or electric motor rotor
• Checking conservation of angular momentum in a collision or coupling (e.g., two disks joining)
• Estimating the momentum stored in a rotating energy storage system
• Converting between rpm and rad/s when you have a datasheet that gives speed in rpm

If you only know the mass and shape of an object, you must first compute its moment of inertia. For a solid cylinder of mass m and radius r rotating about its central axis, I = ½ m r². For a solid sphere, I = (2/5) m r². This calculator does not compute I from mass and shape – it expects you to provide I directly.

How to Read the Result

The angular momentum value tells you how much “rotational oomph� the object has. A larger L means it is harder to stop or change its rotation (if no external torque acts). Units are kg·m²/s in the SI system. In US customary units you might see lb·ft·s (pound-foot-second), but the calculator uses SI because that is standard in physics and engineering.

If you input ω in rpm, the first output (using rad/s) will be correct only if you convert rpm to rad/s first. The calculator’s second output gives you the result directly in the rpm-based unit, so you can compare. For example, if you enter I = 2.5 and ω = 60 rpm, the first output would be 2.5 × (60×2π/60) = 2.5 × 6.283 = 15.71 kg·m²/s, and the second output would be 2.5 × 60 = 150.0 kg·m²·rpm. Both represent the same angular momentum.

Common Mistakes and Unit Mix-Ups

• Using rpm without converting – the formula requires rad/s. Multiply rpm by 2Ï€/60 to get rad/s.
• Confusing moment of inertia with mass – I depends on both mass and how far that mass is from the axis. A light wheel with mass at the rim can have a higher I than a heavier wheel with mass concentrated near the hub.
• Forgetting that angular momentum is a vector – the calculator gives the magnitude along the axis of rotation. If you need direction, use the right-hand rule.
• Using degrees instead of radians – the formula works with radians. If your angular velocity is in degrees per second, convert by multiplying by Ï€/180.

Limits of the Formula

The simple L = Iω applies to rigid bodies rotating about a fixed principal axis. It breaks down if the axis is not fixed (e.g., a spinning top that precesses), if the object deforms while rotating, or if you need relativistic speeds. For a particle moving in a straight line but not through the origin, angular momentum is r × p, not Iω. This calculator is for the common rigid-body case.

Also note that moment of inertia is always relative to a specific axis. The same object has different I values for different axes. Make sure you use the I that matches the axis you are analyzing.

Frequently Asked Questions

What is angular momentum in simple words?

Angular momentum is the rotational equivalent of linear momentum. It measures how hard it is to change an object’s rotation. A figure skater pulling in their arms spins faster because their moment of inertia decreases, and angular momentum is conserved (ignoring friction).

Can I use this calculator for a point mass moving in a circle?

Yes. For a point mass m moving at speed v in a circle of radius r, the moment of inertia is I = m r², and angular velocity ω = v/r. Then L = m r² × (v/r) = m v r. You can enter I = m r² directly.

How do I find moment of inertia if I only have mass and shape?

You need to look up the standard formula for the shape. For a solid cylinder about its central axis: I = ½ m r². For a solid sphere: I = (2/5) m r². For a thin rod about its center: I = (1/12) m L². The calculator assumes you already have I.

What is the difference between angular momentum and torque?

Torque is the rate of change of angular momentum: Ï„ = dL/dt. If you apply a net torque, angular momentum changes. If no net external torque acts, angular momentum is conserved. Torque is like force; angular momentum is like momentum.

Does the calculator work for objects rotating at very high speeds?

Yes, as long as the object remains rigid and non-relativistic. At speeds approaching the speed of light, relativistic effects would change the mass distribution, but that is far beyond everyday engineering.

Why is the second output in kg·m²·rpm? That seems like a weird unit.

It is a convenience unit. Since ω in rpm is common in engineering, expressing angular momentum as I × rpm gives a number that is easy to compare without converting. It is not an official SI unit, but it is used informally. The calculator shows both so you can verify your conversion.

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in I | Moment of inertia | kg·m² | 2.5 | 0
in omega | Angular velocity | rad/s | 6.28 | 0
out L | Angular momentum | kg·m²/s | I * omega | 2
out L_rpm | Angular momentum (rev/min basis) | kg·m²·rpm | I * omega * 60 / (2 * pi) | 2