Q.Choose the correct alternatives: (Note: more than one of the given options may be correct.)

The key concept here is understanding the vector relationships between momentum, velocity, angular momentum, and angular velocity, and how scalar or tensor quantities mediate these relationships.

1. Option (A): For a general rotational motion, the angular momentum $\vec{L}$ is related to the angular velocity $\vec{\omega}$ through the inertia tensor $\mathbf{I}$ as $\vec{L} = \mathbf{I}\vec{\omega}$. Since $\mathbf{I}$ is a tensor, $\vec{L}$ is generally not parallel to $\vec{\omega}$ unless $\vec{\omega}$ lies along a principal axis of inertia. Thus, this statement is correct.
2. Option (B): Even for rotation about a fixed axis, if the rotating body is not symmetric with respect to that axis, the angular momentum vector $\vec{L}$ will not be parallel to the angular velocity vector $\vec{\omega}$. Only the component of $\vec{L}$ along the fixed axis is parallel to $\vec{\omega}$. Thus, this statement is incorrect. …

In physics, understanding the vector relationships between quantities is crucial. Vectors have both magnitude and direction. When we say two vectors are "parallel," it means they point in the same direction or exactly opposite directions. This problem tests our understanding of these relationships for both rotational and translational motion.

Let's analyze each option:

1. Option (A): For a general rotational motion, angular momentum $\vec{L}$ and angular velocity $\vec{\omega}$ need not be parallel.

   • Concept: Angular momentum $\vec{L}$ and angular velocity $\vec{\omega}$ are related through the moment of inertia. For a rigid body, this relationship is expressed using the moment of inertia tensor, $I$.
   • Explanation: The angular momentum $\vec{L}$ of a rigid body rotating with angular velocity $\vec{\omega}$ is given by $\vec{L} = I \vec{\omega}$. Here, $I$ is not a simple scalar but a second-rank tensor (a $3 \times 3$ matrix). When a vector is multiplied by a tensor, the resulting vector does not necessarily point in the same direction as the original vector.
   • Example: Consider a rectangular block rotating about an axis that is not aligned with any of its symmetry axes. Even if the angular velocity $\vec{\omega}$ is constant in direction, the angular momentum $\vec{L}$ will generally not be parallel to $\vec{\omega}$. In fact, $\vec{L}$ might even precess around $\vec{\omega}$ if the body is not rotating about a principal axis.
   • Conclusion: This statement is correct. $\vec{L}$ and $\vec{\omega}$ are generally not parallel for arbitrary rotational motion. They are parallel only if the axis of rotation is a principal axis of inertia for the body.

   The relationship between angular momentum $\vec{L}$ and angular velocity $\vec{\omega}$ for a rigid body is given by:

   $\vec{L} = I \vec{\omega}$

   where $I$ is the moment of inertia tensor.

2. Option (B): For a rotational motion about a fixed axis, angular momentum $\vec{L}$ and angular velocity $\vec{\omega}$ are always parallel.

   • Concept: This statement is a common point of confusion. While the angular velocity $\vec{\omega}$ is indeed fixed in direction along the axis of rotation, the angular momentum $\vec{L}$ is not necessarily parallel to it.
   • Explanation: As discussed in Option (A), $\vec{L} = I \vec{\omega}$. Even if the axis of rotation is fixed (meaning $\vec{\omega}$ has a constant direction), $\vec{L}$ will only be parallel to $\vec{\omega}$ if the fixed axis of rotation is a principal axis of inertia for the body. A principal axis is an axis about which the products of inertia ($I_{xy}, I_{yz}, I_{zx}$) are zero. If the axis is not a principal axis, then even with a fixed $\vec{\omega}$, the angular momentum $\vec{L}$ will have components perpendicular to $\vec{\omega}$. These perpendicular components mean that $\vec{L}$ is not parallel to $\vec{\omega}$.
   • Example: Imagine a bicycle wheel with a small weight attached to its rim, rotating about its axle (a fixed axis). The axle is no longer a principal axis due to the added weight. As the wheel spins, the angular momentum vector $\vec{L}$ will wobble or precess around the fixed angular velocity vector $\vec{\omega}$, indicating they are not parallel.
   • Conclusion: This statement is incorrect. $\vec{L}$ and $\vec{\omega}$ are not always parallel for rotational motion about a fixed axis; they are parallel only if the fixed axis is a principal axis.
   Watch out

   A common misconception is that for fixed-axis rotation, $\vec{L}$ and $\vec{\omega}$ must be parallel. This is only true if the fixed axis of rotation is a principal axis of inertia. If it's not, $\vec{L}$ will generally not be parallel to $\vec{\omega}$ and may even precess around it.

3. Option (C): For a general translational motion, momentum $\vec{p}$ and velocity $\vec{v}$ are always parallel.

   • Concept: Momentum is defined as the product of mass and velocity. …

Concept: Vector Relationships — $\vec L=\mathbf I\vec\omega$ (tensor) vs $\vec p=m\vec v$ (scalar)

Step 1: Evaluate option (A)

For general rotational motion, $\vec L = \mathbf{I}\vec\omega$ where $\mathbf I$ is the moment-of-inertia tensor, not a scalar. Multiplying $\vec\omega$ by a tensor need not preserve direction, so $\vec L$ need not be parallel to $\vec\omega$. True.

Step 2: Evaluate option (B)

Even about a fixed axis, if that axis is not a principal axis of the body, $\vec L$ has components perpendicular to $\vec\omega$ (only the along-axis component matches). So $\vec L$ and $\vec\omega$ are NOT always parallel. False.

Step 3: Evaluate option (C) …