Question

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 $\text{A}$ is $L_A$ which lies in the positive $z-$direction and the angular momentum of $M$ about point $\text{B}$ is $L_B$. The correct statement for this system is :

• A. $L_A$ and $L_B$ are both constant in magnitude and direction.
• B. $L_B$ is constant, both in magnitude and direction.
• C. $L_A$ is constant, both in magnitude and direction.
• D. $L_A$ is constant in direction with varying magnitude.

Answer 1

The correct option is C $L_A$ is constant, both in magnitude and direction.

About point $\text{A}$ :

Angular momentum is given by,
$\overrightarrow{L}=\overrightarrow{r}\times \overrightarrow{p}=M(\overrightarrow{r}\times \overrightarrow{v})$
So,
$L_A=Mvr\sin \theta =Mvr\sin {90}^{\circ }=Mvr$

Speed $v$ is constant as $\omega$ and $r$ are constant, $\because v=\omega r$.

Also, mass $M$ is constant.

Therefore, $L_A$ is constant in magnitude.

The direction of velocity $\overrightarrow{v}$ is always tangential to the path.

$\overrightarrow{r}\times \overrightarrow{v}$ points in vertically upwards direction.
So, the direction of $\overrightarrow{L_A}$ about $\text{A}$ is also constant.

Hence, $L_A$ is constant, both in magnitude and direction.

About point $\text{B}$ :
Here, $L_B=Mvl\sin \theta$
The angle between $\overrightarrow{l}$ and $\overrightarrow{v}$ is always ${90}^{\circ }$, so the magnitude is constant.
However, the direction of $\overrightarrow{L_B}$ changes continuously as the mass rotate.

Hence, option $\left(C\right)$ is the correct answer.