A mass $$M$$ hangs on a massless rod of length $$l$$ which rotates at a constant angular frequency. The mass $$M$$ moves with steady speed in a circular path of constant radius. Assume that the system is in steady circular motion with constant angular velocity $$\omega$$. The angular momentum of $$M$$ about point $$A$$ is $$L_A$$ which lies in the positive $$z$$ direction and the angular momentum of $$M$$ about $$B$$ is $$L_B$$. The correct statement for this system is:

We need to analyze the characteristics of the angular momentum of a mass $$M$$ about two different reference points, $$A$$ and $$B$$, during steady conical pendulum-like circular motion.

1. Analyze Angular Momentum about Point A ($$\vec{L}_A$$)

Point $$A$$ lies at the center of the circular path of the mass $$M$$ along the vertical axis of rotation ($$z$$-axis).

• The position vector $$\vec{r}_A$$ of the mass relative to point $$A$$ lies entirely within the horizontal plane of rotation.
• The linear velocity vector $$\vec{v}$$ is tangential to the circle at every point.
• By definition, the angular momentum vector is:

  $$\vec{L}_A = \vec{r}_A \times \vec{p} = m(\vec{r}_A \times \vec{v})$$

• Using the right-hand rule, the direction of $$\vec{L}_A$$ points straight up along the positive $$z$$-axis at all times. Since the speed and radius are constant, its magnitude ($$m v r_A$$) and its direction remain completely fixed throughout the motion.

2. Analyze Angular Momentum about Point B ($$\vec{L}_B$$)

Point $$B$$ is located on the vertical rotation axis but at a height above the plane of the circle (the suspension point of the rod).

• The position vector $$\vec{r}_B$$ points from $$B$$ down to the mass $$M$$, tracing out the surface of a cone during motion.
• The angular momentum about $$B$$ is:

  $$\vec{L}_B = m(\vec{r}_B \times \vec{v})$$

• Since $$\vec{r}_B$$ and $$\vec{v}$$ are always perpendicular, the magnitude of $$\vec{L}_B$$ is constant ($$m v l$$).
• However, by the right-hand rule, the vector $$\vec{L}_B$$ is always perpendicular to the rod. As the mass rotates, the vector $$\vec{L}_B$$ sweeps out a cone around the vertical $$z$$-axis. Because its orientation is continuously changing in space, $$\vec{L}_B$$ is not constant in direction.

Conclusion

Therefore, $$\vec{L}_A$$ remains constant both in magnitude and direction, while $$\vec{L}_B$$ changes its direction continuously. This corresponds to the correct choice on the  page:

$$\vec{L}_A$$ is constant, both in magnitude and direction (Option D).