Calculate angular momentum L = Iω for rigid bodies and L = mvr for particles. Explore conservation of angular momentum with before/after scenarios.
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 tool helps students and engineers compute angular momentum, rotational kinetic energy, and the effects of changing geometry on rotation.
Angular momentum calculations require knowing the moment of inertia, which depends on the object shape. This calculator provides 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.
Rigid body: L = Iω, I depends on shape. Particle: L = mvr. Conservation: I₁ω₁ = I₂ω₂. Rotational KE = ½Iω².
Result: 54 kg·m²/s
A 60 kg figure skater modeled as a solid cylinder (R = 0.3 m) spinning at 20 rad/s has I = ½(60)(0.3²) = 2.7 kg·m² and L = 2.7 × 20 = 54 kg·m²/s.
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.
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.
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.
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.
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.
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.
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.
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.
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.