1. Interpret the motion

A mass $M$ is attached to a massless rod of length $l$ and rotates with constant angular speed $\omega$ in a horizontal circle of constant radius.

This is the standard conical pendulum / steady circular motion situation.

Let the center of the circular path be point $A$. Since the problem states that the angular momentum about $A$ is along the positive $z$-direction, the motion is in the horizontal plane and the axis of rotation is the $z$-axis.

2. Angular momentum about point $A$

For circular motion about the center $A$,

where $\vec r_A$ is the position vector of the mass from $A$.

Since the particle moves in a circle of constant radius $r$ with constant speed $v$,

which is constant.

Also, $\vec r_A$ and $\vec v$ always lie in the plane of motion, so $\vec L_A$ is perpendicular to that plane, i.e. along the fixed $z$-axis.

Hence:

• magnitude of $\vec L_A$ is constant,
• direction of $\vec L_A$ is constant.

So $\vec L_A$ is constant both in magnitude and direction.

Thus, option A is true.

3. Angular momentum about point $B$

Point $B$ is the suspension point (top end of the rod). The mass moves in a horizontal circle below it.

Angular momentum about $B$ is

where $\vec r_B$ is the vector from $B$ to the mass.

Now, in conical motion:

• $|\vec r_B| = l$ is constant,
• $|\vec v|$ is constant,
• the angle between $\vec r_B$ and $\vec v$ is always $90^\circ$ because velocity is tangent to the circle and perpendicular to the rod's horizontal projection; in fact for the conical pendulum geometry, $\vec r_B$ has a fixed vertical component and rotating horizontal component, while $\vec v$ is horizontal and tangent, making the cross-product magnitude constant.

So the magnitude of $\vec L_B$ is constant.

But its direction is not constant. As the rod rotates, the vector $\vec r_B$ changes direction in space, and hence $\vec r_B \times m\vec v$ also changes direction with time.

Therefore:

• $\vec L_B$ is not constant in direction,
• so it is not constant as a vector.

Thus options B and C are false.

4. Check option D

Option D says both $\vec L_A$ and $\vec L_B$ are constant in magnitude and direction.

Since $\vec L_B$ is not constant in direction, option D is false.

5. Final conclusion

Only option A is correct.

6. Comparison with stored answer

Stored correct answer: A

My derived answer: A

They agree.