Angular momentum represents the rotational analog of linear momentum, quantifying the amount of rotation an object possesses and its resistance to changes in that rotation. This fundamental quantity in classical mechanics governs everything from the spin of electrons to the orbits of galaxies, making it one of the most universally important concepts in physics.
Unlike linear momentum, which describes motion in a straight line, angular momentum characterizes rotational motion about an axis. The concept emerges naturally when analyzing systems that spin, orbit, or otherwise move in curved paths. Its conservation forms one of the foundational principles of physics, explaining phenomena ranging from why ice skaters spin faster when they pull in their arms to why the Earth maintains its rotation for billions of years.
For a point mass: L = mvr
For a rotating body: L = Iω
Where L is angular momentum, measured in kg·m²/s
Variables explained:
For a particle moving in a circular path or any curved trajectory, angular momentum equals the product of its linear momentum (mv) and its perpendicular distance from the reference point or axis. More precisely, angular momentum is the cross product of the position vector and linear momentum: L = r × p.
This vector nature means angular momentum has both magnitude and direction. The direction follows the right-hand rule: curl your fingers in the direction of rotation, and your thumb points along the angular momentum vector. This convention determines whether angular momentum is positive or negative relative to a chosen coordinate system.
For extended objects rotating about an axis, angular momentum depends on the moment of inertia and angular velocity. The moment of inertia replaces mass as the measure of rotational inertia, accounting for how mass distributes relative to the rotation axis. A wheel with mass concentrated at its rim has greater moment of inertia than one with mass near the center, even at equal total mass.
| Shape | Axis | Moment of Inertia |
|---|---|---|
| Point mass | Distance r | I = mr² |
| Solid sphere | Through center | I = (2/5)mr² |
| Hollow sphere | Through center | I = (2/3)mr² |
| Solid cylinder | Central axis | I = (1/2)mr² |
| Thin hoop | Central axis | I = mr² |
| Thin rod | Through center | I = (1/12)mL² |
| Thin rod | Through end | I = (1/3)mL² |
In the absence of external torques, the total angular momentum of a system remains constant. This conservation law ranks among the most important principles in physics, governing behavior from atomic to astronomical scales.
L₁ = L₂
I₁ω₁ = I₂ω₂
When moment of inertia changes, angular velocity must compensate to keep L constant.
The classic demonstration involves an ice skater spinning with arms extended. When they pull their arms close to their body, their moment of inertia decreases. Since angular momentum must remain constant (no external torques), their angular velocity increases proportionally. A skater can increase rotation speed from 2 rev/s to over 6 rev/s simply by changing arm position.
Given:
Calculate final angular velocity:
L = I₁ω₁ = 4.5 × 12.57 = 56.57 kg·m²/s
ω₂ = L/I₂ = 56.57/1.5 = 37.7 rad/s
Result: 37.7 rad/s = 6 rev/s (tripled rotation speed!)
Planets, moons, and satellites possess orbital angular momentum due to their motion around central bodies. For circular orbits, this equals L = mvr, where r is the orbital radius. For elliptical orbits, angular momentum remains constant even as the object speeds up and slows down throughout its orbit.
This conservation explains Kepler's second law: planets sweep equal areas in equal times. As a planet approaches the Sun (smaller r), it must move faster to maintain constant angular momentum. When farther away (larger r), it moves slower.
| Object | Orbital Angular Momentum | Notes |
|---|---|---|
| Earth around Sun | 2.66 × 10⁴⁰ kg·m²/s | Dominates Sun-Earth system |
| Moon around Earth | 2.88 × 10³⁴ kg·m²/s | Slowly increasing |
| ISS around Earth | ~3.2 × 10¹⁵ kg·m²/s | 420 km altitude |
| Electron in H atom | 1.05 × 10⁻³⁴ kg·m²/s | Quantized (ℏ) |
Objects can possess both spin angular momentum (rotation about their own axis) and orbital angular momentum (motion around an external point). Earth demonstrates both: it spins on its axis (creating day and night) while orbiting the Sun (creating years). These two contributions are independent and add vectorially.
The Earth's spin angular momentum measures approximately 7.2 × 10³³ kg·m²/s, much smaller than its orbital angular momentum. This means Earth's orbit around the Sun contains vastly more rotational inertia than Earth's daily spin.
At the quantum level, angular momentum takes only discrete values, fundamentally changing its character. Electron spin, despite its name, represents intrinsic angular momentum that has no classical analog. Electrons possess spin angular momentum of ±ℏ/2, where ℏ (h-bar) is the reduced Planck constant.
Orbital angular momentum in atoms can only take values that are integer multiples of ℏ. This quantization explains atomic structure, chemical bonding, and the stability of matter itself. The total angular momentum of an atom determines its magnetic properties and spectroscopic behavior.
Gyroscopes: These devices exploit angular momentum conservation to maintain orientation in space. A spinning gyroscope resists changes to its rotation axis, enabling navigation systems in aircraft, ships, and spacecraft. The larger the angular momentum, the more stable the gyroscope.
Flywheels: Energy storage systems use massive rotating wheels to store kinetic energy. The stored energy relates directly to angular momentum and moment of inertia. Modern flywheels made from carbon fiber composites can store significant energy for grid stabilization and vehicle applications.
Reaction Wheels: Spacecraft use spinning wheels to control orientation without expending fuel. By changing the wheel's spin rate, the spacecraft rotates in the opposite direction, conserving total angular momentum. The Hubble Space Telescope uses reaction wheels for precise pointing.
Torque (τ) represents the rotational analog of force and equals the rate of change of angular momentum:
τ = dL/dt = Iα
Where α is angular acceleration (rad/s²)
This relationship mirrors Newton's second law (F = ma) for rotational motion. Just as force changes linear momentum, torque changes angular momentum. Zero net torque means constant angular momentum, explaining conservation in isolated systems.
When torque acts perpendicular to angular momentum, the result is precession: a change in the direction of the angular momentum vector rather than its magnitude. A spinning top demonstrates this beautifully. Gravity creates a torque that tries to tip the top over, but instead of falling, the top's axis traces a cone in space.
Earth itself precesses, completing one cycle every 26,000 years. This precession, caused by the gravitational pull of the Sun and Moon on Earth's equatorial bulge, slowly shifts which stars appear in the night sky and will eventually change which star serves as the "North Star."
Moment of inertia (I) measures how mass distributes relative to the rotation axis, analogous to mass in linear motion. Angular momentum (L) measures the amount of rotation, incorporating both the moment of inertia and how fast the object spins. A stationary flywheel has moment of inertia but zero angular momentum.
Cats exploit angular momentum conservation through body contortion. By changing their moment of inertia differently for front and back halves of their body, they can rotate without external torque. This "falling cat problem" fascinated physicists and contributed to understanding of geometric phases in physics.
Earth's angular momentum creates the Coriolis effect, deflecting moving air masses. This deflection causes hurricanes to rotate counterclockwise in the Northern Hemisphere and clockwise in the Southern Hemisphere. The same effect influences ocean currents and even affects artillery trajectories over long distances.
Angular momentum transfers between objects through torques. When you push on a merry-go-round, you transfer angular momentum from your linear motion to the platform's rotation. The Moon transfers angular momentum to Earth's oceans through tidal forces, gradually slowing Earth's rotation while the Moon moves farther away.
Angular momentum uses kg·m²/s in SI units, equivalent to J·s (joule-seconds) or N·m·s. In quantum mechanics, the natural unit is ℏ (h-bar), approximately 1.055 × 10⁻³⁴ J·s. Atomic and molecular angular momenta are typically expressed as multiples of ℏ.
This angular momentum calculator provides comprehensive tools for rotational physics:
Results include unit conversions (rad/s to RPM, SI to common units) for practical applications across physics, engineering, and astronomy.
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