Angular Momentum of a Particle: L = m(r × v)

TITLE: Angular Momentum of a Particle: L = m(r × v) DESCRIPTION: Learn how to calculate the angular momentum of a particle about a point using the formula L = m(r × v), focusing on choosing the correct reference point and vector cross product.

Concept Overview

This question tests the fundamental understanding of angular momentum for a point particle. Angular momentum is a vector quantity that describes the rotational inertia of an object. It is defined as the cross product of the position vector from the reference point to the particle and the linear momentum of the particle. The key to solving such problems lies in correctly identifying the position vector and the velocity vector, and then performing the vector cross product.

Step 1: Recall the definition of angular momentum for a point particle. The angular momentum ($L$) of a particle of mass ($m$) with velocity ($v$) about a point O is given by the vector cross product of the position vector ($r$) from O to the particle and its linear momentum ($p = mv$).

This formula tells us that angular momentum depends on the particle's mass, its velocity, and its position relative to the chosen reference point.

Step 2: Identify the position vector ($\vec{r}$) and the velocity vector ($\vec{v}$). The position vector $\vec{r}$ is drawn from the reference point (the point about which angular momentum is to be found) to the particle. The velocity vector $\vec{v}$ is the instantaneous velocity of the particle. It's crucial to express both vectors in a consistent coordinate system (e.g., Cartesian coordinates). If the reference point is the origin (0,0,0), then $\vec{r}$ is simply the position vector of the particle. If the reference point is not the origin, say at $\vec{r}_{ref}$, then the position vector from the reference point to the particle is $\vec{r} = \vec{r}_{particle} - \vec{r}_{ref}$.

Step 3: Calculate the linear momentum vector ($\vec{p}$). The linear momentum is the product of the particle's mass and its velocity.

Ensure that both $m$ and $\vec{v}$ are correctly identified. If the velocity is given as components, e.g., $\vec{v} = v_x \hat{i} + v_y \hat{j} + v_z \hat{k}$, then $\vec{p} = m(v_x \hat{i} + v_y \hat{j} + v_z \hat{k})$.

Step 4: Compute the vector cross product. The angular momentum is found by taking the cross product of $\vec{r}$ and $\vec{p}$. If $\vec{r} = r_x \hat{i} + r_y \hat{j} + r_z \hat{k}$ and $\vec{p} = p_x \hat{i} + p_y \hat{j} + p_z \hat{k}$, the cross product can be calculated using a determinant:

Expanding this determinant gives:

This will yield the components of the angular momentum vector.

Step 5: Consider the magnitude and direction. The magnitude of the angular momentum is given by $L = |\vec{r}| |\vec{p}| \sin \theta$, where $\theta$ is the angle between $\vec{r}$ and $\vec{p}$. The direction of $\vec{L}$ is perpendicular to the plane formed by $\vec{r}$ and $\vec{p}$, following the right-hand rule. Often, problems simplify if the reference point is chosen strategically, for example, if $\vec{r}$ and $\vec{v}$ are perpendicular, or if the reference point lies on the line of action of the velocity.

Key Takeaways:

• Angular momentum ($\vec{L}$) of a particle is $\vec{r} \times m\vec{v}$.
• The position vector $\vec{r}$ is always measured from the reference point to the particle.
• The cross product calculation is essential for finding both the magnitude and direction of $\vec{L}$.
• Choosing a convenient reference point can simplify the calculation significantly.

Answer: The angular momentum of a particle about a point is found using the formula $\vec{L} = \vec{r} \times m\vec{v}$, where $\vec{r}$ is the position vector from the point to the particle and $\vec{v}$ is the particle's velocity.