Vector Form of Angular Momentum

Angular momentum is a fundamental concept in physics, particularly in the study of rotational motion. It is a vector quantity that represents the rotational inertia and the rotational velocity of an object around a particular axis. The vector form of angular momentum is crucial for understanding the behavior of rotating systems in classical mechanics, as well as in quantum mechanics.

Definition

The angular momentum $\vec{L}$ of a particle with respect to a point O is defined as the cross product of the particle's position vector $\vec{r}$ (from point O to the particle) and its linear momentum $\vec{p}$ (mass times velocity).

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

For a particle of mass $m$ moving with velocity $\vec{v}$, the linear momentum is $\vec{p} = m\vec{v}$, so the angular momentum becomes:

$$\vec{L} = \vec{r} \times m\vec{v}$$

Properties of Angular Momentum

Angular momentum has several important properties:

• Direction: The direction of the angular momentum vector is perpendicular to the plane formed by the position vector and the velocity vector, following the right-hand rule.
• Magnitude: The magnitude of the angular momentum is given by $L = rpv\sin\theta$, where $\theta$ is the angle between $\vec{r}$ and $\vec{v}$.
• Conservation: In a closed system with no external torques, the total angular momentum is conserved.

Angular Momentum for a System of Particles

For a system of particles, the total angular momentum is the vector sum of the angular momenta of the individual particles.

$$\vec{L}_{\text{total}} = \sum_i \vec{L}_i = \sum_i (\vec{r}_i \times m_i\vec{v}_i)$$

Angular Momentum of a Rigid Body

For a rigid body rotating about an axis with angular velocity $\vec{\omega}$, the angular momentum can be expressed as:

$$\vec{L} = I\vec{\omega}$$

where $I$ is the moment of inertia of the body about the axis of rotation.

Conservation of Angular Momentum

The principle of conservation of angular momentum states that if no external torque acts on a system, the total angular momentum of the system remains constant.

$$\frac{d\vec{L}}{dt} = \vec{\tau}_{\text{ext}}$$

If $\vec{\tau}_{\text{ext}} = 0$, then $\frac{d\vec{L}}{dt} = 0$ and $\vec{L}$ is conserved.

Differences and Important Points

Property	Linear Momentum	Angular Momentum
Definition	Product of mass and velocity	Cross product of position vector and linear momentum
Symbol	$\vec{p}$	$\vec{L}$
Units	kg·m/s	kg·m²/s
Conservation	Conserved if no external force	Conserved if no external torque
Direction	Along the direction of velocity	Perpendicular to the plane of rotation

Formulas

Here are some key formulas related to angular momentum:

• Angular momentum of a particle: $\vec{L} = \vec{r} \times m\vec{v}$
• Magnitude of angular momentum: $L = rpv\sin\theta$
• Total angular momentum for a system: $\vec{L}_{\text{total}} = \sum_i (\vec{r}_i \times m_i\vec{v}_i)$
• Angular momentum of a rigid body: $\vec{L} = I\vec{\omega}$
• Conservation of angular momentum: $\frac{d\vec{L}}{dt} = \vec{\tau}_{\text{ext}}$

Examples

Example 1: Single Particle

A particle of mass 2 kg is moving in a circle of radius 3 m with a constant speed of 4 m/s. Calculate the magnitude of its angular momentum about the center of the circle.

Solution:

Since the particle is moving in a circle, the angle $\theta$ between $\vec{r}$ and $\vec{v}$ is 90 degrees, and $\sin\theta = 1$. The magnitude of the angular momentum is:

$$L = rpv\sin\theta = (3\,\text{m})(2\,\text{kg})(4\,\text{m/s})(1) = 24\,\text{kg·m²/s}$$

Example 2: Rigid Body

A solid cylinder of mass 5 kg, radius 0.5 m, and length 1 m is spinning about its central axis at an angular velocity of 6 rad/s. Calculate its angular momentum.

Solution:

First, we calculate the moment of inertia $I$ for a solid cylinder about its central axis:

$$I = \frac{1}{2}mr^2 = \frac{1}{2}(5\,\text{kg})(0.5\,\text{m})^2 = 0.625\,\text{kg·m²}$$

Now, we can find the angular momentum:

$$\vec{L} = I\vec{\omega} = (0.625\,\text{kg·m²})(6\,\text{rad/s}) = 3.75\,\text{kg·m²/s}$$

Understanding the vector form of angular momentum is essential for analyzing rotational motion and is widely applicable in various fields of physics and engineering.