Angular Momentum for Discrete Particles

Angular momentum is a fundamental concept in physics that describes the rotational motion of objects. It is particularly important in the study of discrete particles, such as atoms, molecules, or any small objects that can be treated as point masses. Angular momentum is a vector quantity that represents the product of a particle's rotational inertia and its rotational velocity.

Definition

For a single particle, the angular momentum $\vec{L}$ about a point is defined as the cross product of the particle's position vector $\vec{r}$ relative to that point and its linear momentum $\vec{p}$:

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

where $\vec{r}$ is the position vector from the origin to the particle and $\vec{p} = m\vec{v}$ is the linear momentum of the particle with mass $m$ and velocity $\vec{v}$.

Angular Momentum in Different Systems

Angular momentum can be considered for a single particle or a system of particles. In a system of particles, the total angular momentum is the vector sum of the angular momenta of the individual particles.

Single Particle

For a single particle moving in a plane with a position vector $\vec{r}$ and linear momentum $\vec{p}$, the magnitude of the angular momentum $L$ is given by:

$$ L = r p \sin(\theta) $$

where $\theta$ is the angle between $\vec{r}$ and $\vec{p}$.

System of Particles

For a system of $n$ particles, the total angular momentum $\vec{L}_{\text{total}}$ is the sum of the angular momenta of all the particles:

$$ \vec{L}{\text{total}} = \sum{i=1}^{n} \vec{L}i = \sum{i=1}^{n} \vec{r}_i \times \vec{p}_i $$

Conservation of Angular Momentum

Angular momentum is conserved in a system where there is no external torque. This means that the total angular momentum of a closed system remains constant over time.

$$ \frac{d\vec{L}}{dt} = \vec{\tau}_{\text{external}} = 0 \Rightarrow \vec{L} = \text{constant} $$

where $\vec{\tau}_{\text{external}}$ is the external torque acting on the system.

Differences and Important Points

Aspect	Single Particle	System of Particles
Definition	$\vec{L} = \vec{r} \times \vec{p}$	$\vec{L}_{\text{total}} = \sum \vec{L}_i$
Conservation	Conserved if no external torque	Conserved if no external torque on the system
Dependence on Reference Point	Yes, $\vec{L}$ depends on the choice of origin	Yes, but total $\vec{L}_{\text{total}}$ can be simplified using center of mass
Calculation	Directly from position and momentum	Summation of individual angular momenta

Examples

Example 1: Single Particle

Consider a particle of mass $m$ moving in a circle of radius $r$ with a constant speed $v$. The angular momentum $L$ of the particle about the center of the circle is:

$$ L = mvr $$

since the angle $\theta$ between $\vec{r}$ and $\vec{v}$ is 90 degrees, and $\sin(90^\circ) = 1$.

Example 2: System of Particles

Imagine two particles, each with mass $m$, moving in opposite directions along a line that passes through a point $O$. The particles have equal and opposite velocities $\vec{v}$ and $-\vec{v}$, and they are at equal distances $r$ from point $O$. The total angular momentum about point $O$ is zero because the angular momenta of the two particles cancel each other out.

$$ \vec{L}_{\text{total}} = \vec{r} \times m\vec{v} + (-\vec{r}) \times m(-\vec{v}) = \vec{0} $$

In summary, angular momentum is a crucial concept in understanding rotational motion. It is conserved in the absence of external torques, and its calculation depends on the position and momentum of the particles involved. Understanding angular momentum for discrete particles is essential for analyzing many physical systems, from atomic scales to celestial mechanics.