4.6 Angular Momentum

Angular momentum is the rotational equivalent of linear momentum. Just as linear momentum describes an object's quantity of motion in a straight line, angular momentum describes an object's quantity of rotational motion. It is a fundamental conserved quantity in physics, crucial for understanding the dynamics of everything from spinning ice skaters and planetary orbits to the behaviour of subatomic particles.

Definition in Terms of Linear Momentum

The angular momentum ($\vec{L}$) of a particle about a certain origin is defined as the cross product of its position vector ($\vec{r}$) relative to the origin and its linear momentum vector ($\vec{p}$).

Vector Formula: $\vec{L}=\vec{r}\times \vec{p}$

Where:

• $\vec{r}$ is the position vector from the axis of rotation to the particle.
• $\vec{p}=m\vec{v}$ is the linear momentum of the particle.

Magnitude: The magnitude of the angular momentum is given by: $L=rp\sin \theta =rmv\sin \theta$ where $\theta$ is the angle between the position vector $\vec{r}$ and the linear momentum vector $\vec{p}$.

For Circular Motion: When an object moves in a circle, the velocity is perpendicular to the radius ($\theta =90^{\circ }$, $\sin 90^{\circ }=1$), so: $L=rmv$

Definition for a Rigid Body

For a rigid body rotating about a fixed axis, angular momentum is expressed in terms of moment of inertia ($I$) and angular velocity ($\omega$):

$L=I\omega$

Where:

• $I$ is the moment of inertia — a measure of an object's resistance to changes in rotational motion (analogous to mass in linear motion).
• $\omega$ is the angular velocity — the rate of rotation.

Direction of Angular Momentum

Angular momentum is a vector quantity. Its direction is perpendicular to the plane of rotation and is determined by the right-hand rule:

1. Curl the fingers of your right hand in the direction of the object's rotation.
2. Your extended thumb points in the direction of $\vec{L}$.

Units and Dimensions of Angular Momentum

The SI unit of angular momentum is $\text{kg}\cdot {\text{m}}^2\cdot {\text{s}}^{-1}$, which is equivalent to Joule-second ($\text{J}\cdot \text{s}$).

Dimensions: Since $L=rp$, $[L]=[r][p]=\text{m}\cdot \text{kg}\cdot \text{m}\cdot {\text{s}}^{-1}=[M{L}^2{T}^{-1}]$

Relating the Two Formulas

The two definitions $L=rmv$ and $L=I\omega$ are directly related. For a point mass $m$ rotating in a circle of radius $r$:

1. Start with: $L=rmv$
2. Use $v=r\omega$: $L=rm(r\omega )$
3. Rearrange: $L=(m{r}^2)\omega$
4. Since $I=m{r}^2$ for a point mass: $L=I\omega$

Law of Conservation of Angular Momentum

The Law of Conservation of Angular Momentum states:

$I_1{\omega }_1=I_2{\omega }_2$

Examples:

• Ice skater: Pulling arms inward decreases $I$, so $\omega$ increases — the skater spins faster.
• Gyroscope: Maintains its orientation because $L$ is conserved in the absence of external torque.
• Flywheel: Stores rotational energy; resists changes in $\omega$ due to large $I$.

Relationship Between Torque and Angular Momentum

Analogous to Newton's second law ($F=\Delta p/\Delta t$), the rotational equivalent is: $\tau =\frac{\Delta L}{\Delta t}$ The net external torque equals the rate of change of angular momentum. When $\tau =0$, $\Delta L=0$ and angular momentum is conserved.

Linear vs Rotational Motion — Analogy Table