A particle has a linear momentum p and position vector r. the angular momentum of this particle about the origin will not be zero under the conditions

Options

A. p=0
B. p is perpendicular to r and does not cross r
C. p is anti-parallel to r
D. the particle passes through the origin.

Explanation

To determine the conditions under which the angular momentum \( L \) of a particle about the origin is not zero, we can use the formula for angular momentum:

\[ L = \mathbf{r} \times \mathbf{p} \]

where:
- \( \mathbf{r} \) is the position vector of the particle,
- \( \mathbf{p} \) is the linear momentum of the particle.

### Step 1: Understand the Cross Product
The angular momentum \( L \) is given by the cross product of the position vector \( \mathbf{r} \) and the linear momentum \( \mathbf{p} \). The magnitude of the cross product can be expressed as:

\[ |L| = |\mathbf{r}| |\mathbf{p}| \sin \theta \]

where \( \theta \) is the angle between the vectors \( \mathbf{r} \) and \( \mathbf{p} \).

### Step 2: Analyze the Conditions for Zero Angular Momentum
For the angular momentum \( L \) to be zero, one of the following conditions must be true:
1. \( |\mathbf{p}| = 0 \) (the particle is at rest).
2. \( \sin \theta = 0 \) (the vectors \( \mathbf{r} \) and \( \mathbf{p} \) are parallel or anti-parallel).
3. \( \mathbf{r} = 0 \) (the particle is located at the origin).

### Step 3: Identify Conditions for Non-Zero Angular Momentum
To find when the angular momentum is not zero, we need to ensure that none of the above conditions hold. This leads us to the following conditions:
1. \( |\mathbf{p}| \neq 0 \) (the particle must be moving).
2. \( \sin \theta \neq 0 \) (the vectors must not be parallel or anti-parallel, meaning \( \theta \) should not be 0 or \( \pi \)).
3. \( \mathbf{r} \neq 0 \) (the particle must not be at the origin).

### Step 4: Evaluate the Given Options
Now, let’s evaluate the options based on the conditions we derived:
1. If \( \mathbf{p} = 0 \), then \( L = 0 \).
2. If \( \mathbf{p} \) is perpendicular to \( \mathbf{r} \) (i.e., \( \theta = 90^\circ \)), then \( L \) is not zero.
3. If \( \mathbf{p} \) is anti-parallel to \( \mathbf{r} \) (i.e., \( \theta = 180^\circ \)), then \( L = 0 \).
4. If the particle passes through the origin, then \( \mathbf{r} = 0 \), and thus \( L = 0 \).

### Conclusion
From the analysis, the angular momentum of the particle about the origin will not be zero under the condition that the linear momentum \( \mathbf{p} \) is perpendicular to the position vector \( \mathbf{r} \) (Option 2).

### Final Answer
The angular momentum of the particle about the origin will not be zero under the condition that \( \mathbf{p} \) is perpendicular to \( \mathbf{r} \).

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