Question

A particle with position vector $\vec{r}$ has a linear momentum $\vec{P}$. Which one of the following statements is true in respect of its angular momentum 'L' about the origin?

• A. $\vec{L}$ acts along $\vec{P}$.
• B. $L$ is maximum when $\vec{P}$ is perpendicular to $\vec{r}$.
• C. $\vec{L}$ acts along $\vec{r}$.
• D. $L$ is maximum when $\vec{P}$ and $\vec{r}$ are parallel.

Hint:

Think of this in terms of torque or rotation shortcuts: a force or momentum creates the most rotation when it is applied perpendicularly to the lever arm. When they are parallel ($\theta = 0^\circ$), the cross product vanishes entirely ($L = 0$).

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
The problem asks us to evaluate the properties of the angular momentum vector $\vec{L}$ of a moving point particle, defined relative to the coordinate origin in terms of its position vector $\vec{r}$ and linear momentum vector $\vec{P}$.

Step 2: Key Formula or Approach:
The fundamental vector cross-product definition for angular momentum is: $$\vec{L} = \vec{r} \times \vec{P}$$ The scalar magnitude version of this cross-product expression is: $$L = r P \sin\theta$$ where $\theta$ represents the angle between vectors $\vec{r}$ and $\vec{P}$.

Step 3: Detailed Explanation:
Let's analyze the properties of this cross-product system: 1.

Direction: By the definition of a cross product, the resulting vector $\vec{L}$ must be simultaneously perpendicular to both originating vectors, $\vec{r}$ and $\vec{P}$. Therefore, statements (A) and (C) are incorrect. 2.

Magnitude Optimization: The magnitude expression $L = r P \sin\theta$ depends directly on the value of $\sin\theta$. The sine function reaches its maximum value of 1 when the angle is exactly $90^\circ$: $$\sin\theta = 1 \implies \theta = 90^\circ \quad (\vec{r} \perp \vec{P})$$ This means the angular momentum reaches its maximum value ($L_{\text{max}} = rP$) when the linear momentum vector is perpendicular to the position vector. This matches statement (B).

Step 4: Final Answer:
The correct statement is that $L$ is maximum when $\vec{P}$ is perpendicular to $\vec{r}$, which corresponds to option (B).