Angular Momentum

Equivalent forms

\vec{L} = \vec{r} \times \vec{p}

L = mvr\sin\theta

The conserved quantity behind spinning skaters, planetary orbits, and electron spin.

Symbol	Name	SI unit	Dimension	Range
$L$	Angular Momentumoutput Rotational analogue of linear momentum	kg·m²/s	ML^2T^-1	0 – 1000
$I$	Moment of Inertia Rotational inertia about the axis	kg·m²	ML^2	0.1 – 20
$ω$	Angular Velocity Rate of angular rotation	rad/s	T^-1	0 – 20

Unit systems

Symbol	SI	CGS	Imperial
$L$	kg·m²/s	g·cm²/s	slug·ft²/s
$I$	kg·m²	g·cm²	slug·ft²
$ω$	rad/s	rad/s	rad/s

Where it holds

Valid for rigid rotation about a principal axis. For arbitrary axes,

L = I\omega

becomes a tensor equation. Conservation holds when net external torque is zero.

Dimensional analysis

L \to [ML^{2}T^{-1}]

I\cdot \omega \to [ML^{2}]\cdot [T^{-1}] = [ML^{2}T^{-1}]

Both sides are

kg\cdot m^{2}/s

.

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Discovery

Leonhard Euler / Johannes Kepler · 1765

Kepler's second law (equal areas in equal times) is conservation of L. Euler formalized rigid-body angular momentum tensor in the 18th century.

Try this

Why does a figure skater spin faster when she pulls her arms in?

A skater has moment of inertia 4 kg·m² with arms out, spinning at 2 rad/s. With arms tucked, I drops to 1.2 kg·m². Find her new angular velocity using L = Iω conservation.

Research status: stable

Real-world applications

Figure skating and gymnastics
Gyroscopes in inertial navigation
Neutron star spin-up after collapse
Helicopter and drone stabilization

Common misconceptions

Angular momentum does not require a 'spinning' object — orbiting point particles also carry L
L is conserved only when net external torque is zero, not when internal forces redistribute mass
Quantum spin has units of ħ but is not literal rotation

Experimental verification

Spinning bicycle wheel demonstrations, gyroscope precession measurements, and Kepler's areal-velocity law from observational astronomy all confirm L conservation.

Derivation

For a point particle:

L = r \times p = r \times (mv)

.

For a rigid body summed over particles,

L = (\Sigma m

ᵢrᵢ

^{2})\omega = I\omega

about the principal axis.

Newton's second law for rotation:

\tau = dL/dt

.

Limiting cases

\omega \to 0

⟶

L \to 0

No rotation, no angular momentum.

I \to 0

⟶

L \to 0

Point particle on the axis carries no spin angular momentum.

External torque

= 0

⟶

L =

constConservation of angular momentum — the law behind skater spins and planetary orbits.

What if…

What if I doubles?

For fixed $\omega$ , L doubles. Under conservation, doubling I forces $\omega to$ halve.

What if a neutron star forms?

A $10^{6} km$ star collapsing to 10 km shrinks $I by \sim 10^{10}$ . Spin rate increases by the same factor — typical pulsars rotate $\sim 1000\times per$ second.

What if you sit on a spinning chair holding weights?

Pulling weights inward lowers I, spinning you faster. Pushing outward slows you down — internal forces redistribute mass without changing L.

1

Figure skater pulling arms in

Given ·

I1:: 4
ω1:: 2
I2:: 1.2

Find ·

\omega 2

Steps

Conservation: $I_{1}\cdot \omega 1 = I_{2}\cdot \omega 2$
$4 \times 2 = 1.2 \times \omega 2$
$\omega 2 = 8 / 1.2 \approx 6.67\,\mathrm{rad/s}$

Result ·

\omega 2 = 6.67\,\mathrm{rad/s}

— over

3\times

faster spin

2

Spinning bicycle wheel

Given ·

I:: 0.15
ω:: 30

Find · L

Steps

Wheel moment of inertia $\approx 0.15 kg\cdot m^{2} for$ a typical 26-inch wheel
Spin at $30\,\mathrm{rad/s} \approx 286\,\mathrm{RPM}$
$L = 0.15 \times 30 = 4.5 kg\cdot m^{2}/s$ — strong enough to feel the gyroscopic torque when tilting the axle

Result ·

L = 4.5 kg\cdot m^{2}/s

Torque→Rotational Kinetic Energy→Centripetal Acceleration→