Angular Momentum Calculator

About the Angular Momentum Calculator

Angular momentum is the rotational equivalent of linear momentum and is one of the most fundamental conserved quantities in physics. For a rigid body, L = Iω, where I is the moment of inertia and ω is the angular velocity. For a point particle, L = mvr, where m is mass, v is velocity, and r is the distance from the rotation axis.

This calculator supports three modes: rigid body rotation (with seven common shapes for automatic moment of inertia calculation), point particle orbital motion, and conservation of angular momentum scenarios. The conservation mode demonstrates the famous figure skater effect — when a spinning object reduces its moment of inertia, it must spin faster to conserve angular momentum.

Angular momentum conservation governs phenomena from pirouettes to planetary orbits, from neutron star formation to the stability of bicycle wheels. This calculator helps students and engineers compute angular momentum, rotational kinetic energy, and the effects of changing geometry on rotation.

When This Page Helps

Angular momentum calculations require knowing the moment of inertia, which depends on the object shape. It gives seven common shapes with automatic I computation, eliminating the need to look up formulas.

The conservation mode is particularly valuable — it directly shows how changing the moment of inertia affects angular velocity and kinetic energy, making abstract conservation laws tangible and quantitative.

How to Use the Inputs

1. Select a mode: rigid body (L = Iω), point particle (L = mvr), or conservation.
2. For rigid bodies, choose the object shape to auto-calculate moment of inertia.
3. Enter mass, radius/length, and angular velocity as required.
4. For conservation mode, enter initial I and ω, then the final I.
5. Use presets to explore real-world examples like a figure skater or Earth orbit.
6. Read the angular momentum, kinetic energy, and related outputs.

Example Calculation

Tips & Best Practices

• For compound objects, add the individual moments of inertia (they are additive).
• The parallel axis theorem says I = I_cm + md² for an axis offset by distance d from the center of mass.
• In conservation problems, watch the kinetic energy — it changes even though angular momentum does not.
• Angular momentum is a vector — its direction is along the rotation axis by the right-hand rule.
• For point particles, L = mvr only works when v is perpendicular to r. In general, L = r × p.

Conservation of Angular Momentum

Angular momentum conservation is one of the most powerful principles in physics. When no external torques act on a system, the total angular momentum remains constant. This principle explains why planets orbit the Sun in stable ellipses, why pulsars spin at incredible rates, and why a cat can twist to land on its feet.

From Ice Rinks to Neutron Stars

When a massive star collapses into a neutron star, its radius shrinks by a factor of ~10,000. Since angular momentum is conserved (I₁ω₁ = I₂ω₂) and I ∝ R², the angular velocity increases by ~10⁸. A star rotating once per month can become a pulsar spinning hundreds of times per second.

Quantum Angular Momentum

In quantum mechanics, angular momentum is quantized — it can only take discrete values. Orbital angular momentum L = √(l(l+1))ℏ where l is a non-negative integer. Spin angular momentum has half-integer values for fermions (electrons, protons). This quantization underpins the periodic table, chemical bonding, and magnetic properties of materials.

Frequently Asked Questions

• Why does a figure skater spin faster when pulling arms in?

  When the skater pulls their arms in, their moment of inertia decreases. Since angular momentum (L = Iω) is conserved, ω must increase to compensate. A skater can triple their spin rate this way.

• Is angular momentum always conserved?

  Angular momentum is conserved when no external torque acts on the system. In practice, friction provides a small external torque, so angular momentum is approximately conserved over short times.

• What is moment of inertia?

  Moment of inertia (I) is the rotational equivalent of mass. It depends on both the object's mass and how that mass is distributed relative to the rotation axis. Mass farther from the axis contributes more to I.

• How does kinetic energy change in conservation problems?

  While angular momentum is conserved, kinetic energy is NOT necessarily conserved. When a skater pulls arms in, KE increases — the extra energy comes from the muscular work done against centrifugal force.

• What is the angular momentum of the Earth?

  Earth's orbital angular momentum around the Sun is about 2.66 × 10⁴⁰ kg·m²/s, while its spin angular momentum is about 7.07 × 10³³ kg·m²/s.

• How is angular momentum related to gyroscopic stability?

  A spinning wheel resists changes to its rotation axis due to angular momentum conservation. This gyroscopic effect stabilizes bicycles, helps navigation instruments, and makes tops stand upright.