A particle with the position vector r has linear momentum p. Which of the following statements is true is respect of its angular momentum L about the origin ?

Options

A. L acts along p
B. L acts along r
C. L is maximum when p and r are parallel
D. L is maximum when p is perpendicular to r

Explanation

To solve the problem regarding the angular momentum \( L \) of a particle with position vector \( r \) and linear momentum \( p \), we can follow these steps:

### Step-by-Step Solution:

1. **Understanding Angular Momentum**:
The angular momentum \( L \) of a particle about the origin is given by the cross product of the position vector \( r \) and the linear momentum \( p \):
\[ L = r \times p \]

2. **Properties of the Cross Product**:
The magnitude of the angular momentum can be expressed as:
\[ |L| = |r| |p| \sin \theta \]
where \( \theta \) is the angle between the vectors \( r \) and \( p \).

3. **Direction of Angular Momentum**:
The direction of the angular momentum vector \( L \) is given by the right-hand rule, which indicates that \( L \) is perpendicular to both \( r \) and \( p \). Therefore, options stating that \( L \) acts along \( p \) or \( r \) are incorrect.

4. **Maximizing Angular Momentum**:
The magnitude of \( L \) is maximized when \( \sin \theta \) is maximized. The maximum value of \( \sin \theta \) is 1, which occurs when \( \theta = 90^\circ \) (or \( \frac{\pi}{2} \) radians). This means that the vectors \( r \) and \( p \) must be perpendicular to each other for \( L \) to be maximized.

5. **Evaluating the Options**:
- **Option 1**: \( L \) acts along \( p \) → Incorrect (as \( L \) is perpendicular to \( p \)).
- **Option 2**: \( L \) acts along \( r \) → Incorrect (as \( L \) is perpendicular to \( r \)).
- **Option 3**: \( L \) is maximum when \( p \) and \( r \) are parallel → Incorrect (as this would make \( \theta = 0 \) and \( L = 0 \)).
- **Option 4**: \( L \) is maximum when \( p \) is perpendicular to \( r \) → Correct (as this gives \( \theta = 90^\circ \)).

6. **Conclusion**:
The correct answer is that the angular momentum \( L \) is maximum when the linear momentum \( p \) is perpendicular to the position vector \( r \).

### Final Answer:
**The correct statement is Option 4: \( L \) is maximum when \( p \) is perpendicular to \( r \).**