The angular momentum is?

Options

A. scalar quantity
B. in a polar direction
C. along the axis
D. all of these

Answer

To solve the question regarding the nature of angular momentum, we will go through the definition, properties, and characteristics of angular momentum step by step.

### Step-by-Step Solution:

1. **Definition of Angular Momentum**:
Angular momentum (L) is defined as the moment of linear momentum about a point. It can be expressed mathematically as:
\[ \mathbf{L} = \mathbf{R} \times \mathbf{P} \]
where \(\mathbf{R}\) is the position vector from the point of rotation to the particle, and \(\mathbf{P}\) is the linear momentum of the particle.

2. **Expression for Linear Momentum**:
The linear momentum (\(\mathbf{P}\)) of a particle is given by:
\[ \mathbf{P} = m \mathbf{v} \]
where \(m\) is the mass of the particle and \(\mathbf{v}\) is its velocity. Thus, we can rewrite angular momentum as:
\[ \mathbf{L} = \mathbf{R} \times (m \mathbf{v}) = m (\mathbf{R} \times \mathbf{v}) \]

3. **Understanding the Cross Product**:
The cross product \(\mathbf{R} \times \mathbf{v}\) gives a vector that is perpendicular to the plane formed by \(\mathbf{R}\) and \(\mathbf{v}\). The magnitude of this vector is given by:
\[ |\mathbf{L}| = m R v \sin(\theta) \]
where \(\theta\) is the angle between the vectors \(\mathbf{R}\) and \(\mathbf{v}\).

4. **Analyzing Different Cases**:
- **Radial Velocity**: If the velocity is radial (along the line of \(\mathbf{R}\)), then \(\theta = 0^\circ\) or \(180^\circ\), and \(\sin(\theta) = 0\). Hence, the angular momentum \(L = 0\).
- **Transverse Velocity**: If the velocity is perpendicular to \(\mathbf{R}\) (transverse), then \(\theta = 90^\circ\), and \(\sin(90^\circ) = 1\). Thus, the angular momentum becomes:
\[ |\mathbf{L}| = m R v \]

5. **Direction of Angular Momentum**:
The direction of angular momentum is determined by the right-hand rule applied to the cross product. If you point your fingers in the direction of \(\mathbf{R}\) and curl them towards \(\mathbf{v}\), your thumb will point in the direction of \(\mathbf{L}\). This direction is axial, meaning it is along the axis of rotation.

6. **Conclusion**:
From the analysis, we can conclude that angular momentum is a vector quantity, has a direction along the axis of rotation, and is not a scalar quantity. Therefore, the correct answer to the question is that angular momentum is along the axis.

### Final Answer:
Angular momentum is a vector quantity and is directed along the axis of rotation.

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