Angular Momentum Calculator
Calculate angular momentum from moment of inertia and angular velocity. Angular momentum is the rotational analog of linear momentum and is conserved in isolated systems.
Angular Momentum Conservation
Conservation of angular momentum is one of the deepest principles in physics, arising from the rotational symmetry of physical laws. In any system where no external torque acts, the total angular momentum remains constant. This principle governs phenomena from atomic electron orbitals to the rotation of galaxies.
The classic demonstration involves a person sitting on a freely rotating stool holding weights with arms extended. When they pull the weights inward, the moment of inertia decreases and they spin faster. Extending the arms again slows the spin. At no point does an external torque act on the system, so the product Iω stays constant throughout.
Planetary orbits provide a celestial example. As a planet moves closer to the Sun in its elliptical orbit, it speeds up, and as it moves farther away, it slows down. This is Kepler's second law (equal areas in equal times), which is a direct consequence of angular momentum conservation under the central gravitational force that produces no torque about the Sun.
Angular Momentum in Astrophysics
Stars form from slowly rotating gas clouds that collapse under gravity. As the cloud contracts, its moment of inertia shrinks enormously, and conservation of angular momentum forces the rotation rate to increase dramatically. This is why young stars spin rapidly and why collapsing stellar cores form rapidly spinning neutron stars called pulsars, some rotating over 700 times per second.
Accretion disks around black holes and young stars exist because of angular momentum. Infalling material cannot simply plunge straight in because it carries angular momentum. Instead, it spirals inward, forming a flat disk. Friction within the disk gradually transfers angular momentum outward, allowing material to slowly migrate inward and eventually fall onto the central object.
The angular momentum of the solar system is distributed surprisingly: the Sun holds 99.86% of the mass but only about 0.5% of the angular momentum. Jupiter alone carries about 60% of the total angular momentum. This distribution is a clue to how the solar system formed and has puzzled astronomers for centuries.
Angular Momentum in Quantum Mechanics
At the quantum level, angular momentum is quantized. Electrons in atoms can only have specific angular momentum values, which determine the shapes of atomic orbitals and the structure of the periodic table. The orbital angular momentum quantum number l restricts angular momentum to values of √(l(l+1))ħ, where l is a non-negative integer.
Spin is an intrinsic form of angular momentum that has no classical analog. Electrons, protons, and other fundamental particles possess spin angular momentum even though they are not literally spinning. Electron spin takes values of +½ħ or -½ħ, and this two-valuedness underlies the Pauli exclusion principle that governs chemical bonding and the stability of matter.
Nuclear magnetic resonance (NMR) and MRI imaging exploit the angular momentum of atomic nuclei. In a magnetic field, nuclei with spin precess around the field direction at a frequency determined by their angular momentum. Radio-frequency pulses tip the precessing nuclei, and the signals they emit as they relax reveal information about molecular structure in chemistry or tissue properties in medical imaging.
Frequently Asked Questions
What is angular momentum?
Angular momentum (L) is the rotational equivalent of linear momentum. For a rigid body, L = Iω, where I is moment of inertia and ω is angular velocity. For a point mass, L = mvr, where r is the distance from the rotation axis. The SI unit is kg·m²/s.
Is angular momentum conserved?
Yes, angular momentum is conserved when no external torque acts on a system. This is why planets maintain their orbits, why spinning tops stay upright, and why ice skaters spin faster when they pull their arms in. It is one of the most fundamental conservation laws in physics.
How does a figure skater spin faster?
When a skater pulls arms inward, moment of inertia decreases. Since angular momentum L = Iω must remain constant (no external torque), angular velocity ω must increase. Reducing I by half doubles ω. This is the most famous everyday demonstration of angular momentum conservation.
What is the relationship between torque and angular momentum?
Torque equals the rate of change of angular momentum: τ = dL/dt. This is the rotational analog of F = dp/dt. When net torque is zero, angular momentum is constant. When a net torque acts, it changes the angular momentum at a rate equal to the torque magnitude.
What are the units of angular momentum?
The SI unit is kg·m²/s, equivalent to N·m·s or J·s. In quantum mechanics, angular momentum is quantized in multiples of ħ (h-bar), the reduced Planck constant, approximately 1.055 × 10⁻³⁴ J·s.