Angular Momentum
Calculate angular momentum using L = I × ω for rotating rigid bodies or L = m × v × r for particles in circular motion. Free online physics calculator with instant kg·m²/s results, step-by-step breakdowns, and interactive charts for rotational motion problems.
About This Calculator
The Angular Momentum Calculator helps students, engineers, and physics enthusiasts compute angular momentum (rotational momentum) for two common scenarios: rigid bodies rotating about a fixed axis and point masses moving in circular paths. Angular momentum is a conserved vector quantity that plays a crucial role in mechanics, astrophysics, quantum mechanics, and engineering.
The calculator supports two formulas: L = I × ω for rigid bodies (where I is moment of inertia in kg·m² and ω is angular velocity in rad/s) and L = m × v × r for point masses (where m is mass in kg, v is linear velocity in m/s, and r is the radius of the circular path in meters). Results are displayed in the SI unit kg·m²/s with a detailed breakdown of each input component and interactive bar or pie charts for visual analysis.
Key Concepts
Angular momentum is the rotational analogue of linear momentum. It depends not only on mass and velocity but also on how the mass is distributed relative to the axis of rotation (described by the moment of inertia). The direction of angular momentum follows the right-hand rule: when fingers curl in the direction of rotation, the thumb points in the direction of the angular momentum vector, perpendicular to the plane of rotation.
Conservation of Angular Momentum
The law of conservation of angular momentum states that the total angular momentum of a closed system remains constant in the absence of external torque. This principle explains diverse phenomena: ice skaters spinning faster by pulling their arms in, the Moon's tidal locking with Earth, neutron stars spinning at incredible speeds after stellar collapse, the stability of gyroscopes, and planetary orbits following Kepler's law of equal areas. The conservation law is fundamental to understanding central force motion and is one of the three great conservation laws (energy, linear momentum, and angular momentum) that govern classical and quantum physics.
Frequently Asked Questions
What is angular momentum?
Angular momentum, also known as rotational momentum or moment of momentum, is the rotational equivalent of linear momentum. It measures how much rotational motion an object has around an axis. For a rigid body spinning on its axis, angular momentum L = I×ω. For a particle moving in a circular path, L = m×v×r.
How do I calculate angular momentum for a rigid body?
For a rigid body rotating about a fixed axis, use L = I × ω, where I is the moment of inertia in kg·m² and ω is the angular velocity in rad/s. Select the Rigid Body mode in the calculator, enter the moment of inertia and angular velocity, and click Calculate to get the angular momentum in kg·m²/s.
How do I calculate angular momentum for a point mass?
For a point mass moving in a circular path, use L = m × v × r, where m is the mass in kg, v is the linear velocity in m/s, and r is the radius of the circular path in meters. Select the Point Mass mode in the calculator, enter the three values, and click Calculate to get the angular momentum.
What is the conservation of angular momentum?
The law of conservation of angular momentum states that if no external torque acts on a system, its total angular momentum remains constant. This explains why an ice skater spins faster when pulling their arms in (reducing moment of inertia causes angular velocity to increase to keep L constant), and why a spinning gyroscope maintains its orientation.
What are the units of angular momentum?
Angular momentum is measured in kg·m²/s (kilogram meters squared per second) in the SI system. Its dimensions are ML²T⁻¹. In atomic physics, angular momentum is often expressed in units of ħ (reduced Planck constant), where 1 ħ = 1.0546 × 10⁻³⁴ J·s.
What is the difference between spin and orbital angular momentum?
Spin angular momentum is the angular momentum of an object about its own center of mass (e.g., Earth rotating on its axis). Orbital angular momentum is the angular momentum of the center of mass about an external point (e.g., Earth orbiting the Sun). The total angular momentum of a system is the vector sum of spin and orbital angular momentum.
Where is angular momentum used in real-world applications?
Angular momentum is fundamental in many fields: satellite orbital mechanics (central force motion), gyroscopes and navigation systems, ice skating and diving spins, bowling ball motion, planetary orbits (Kepler's law of areas), quantum mechanics (electron spin and orbital angular momentum), astrophysics (black hole rotation), and electromagnetic theory (angular momentum of electromagnetic fields).
How does torque relate to angular momentum?
Torque is the rate of change of angular momentum, just as force is the rate of change of linear momentum. Mathematically, τ = dL/dt. If a net torque acts on a system, its angular momentum changes. When no external torque acts (isolated system), angular momentum is conserved. This relationship is the rotational analogue of Newton's Second Law F = dp/dt.