Question

A small ball of mass m suspended from the ceiling at a point O by a thread of length moves along a horizontal circle with a constant angular velocity ω. Which of the followings is /are correct?

• A. angular momentum is constant about O
• B. angular momentum is constant about C
• C. vertical component of angular momentum about O is constant
• D. Magnitude of angular momentum about O is constant

Answer 1

Answer: (B), (C), (D)

Magnitude of angular momentum about O is constant

(a) ${\overrightarrow{L}}_0$ obviously varies as direction of both $\overrightarrow{r}$ and $\overrightarrow{P}$ keep changing

Example

(b) ${\overrightarrow{L}}_c={\overrightarrow{r}}_c\times \overrightarrow{P}$

$|{\overrightarrow{r}}_c|$ is constant

$|\overrightarrow{P}|$ is constant

and they are always perpendicular

$\therefore \theta ={90}^{\circ }$

$\therefore |{\overrightarrow{L}}_c|=|{\overrightarrow{r}}_c||{\overrightarrow{P}}_c|sin\theta =constant$

And also , direction of $\overrightarrow{L}$ will always be upward

Example

(c) Mangnitude of ${\overrightarrow{L}}_0$ will not change as mangnitude of $|{\overrightarrow{r}}_0|$ and $|\overrightarrow{P}|$ will not change the angle that ${\overrightarrow{L}}_0$

$\therefore$ The vertical component of ${\overrightarrow{L}}_0$ will be constant.

(d) $|{\overrightarrow{L}}_0|=|{\overrightarrow{r}}_0||\overrightarrow{P}|sin\theta$

$|{\overrightarrow{r}}_0|$ is constant

$|\overrightarrow{P}|$ is constant

$\theta$ will always be ${90}^{\circ }$

$\therefore |{\overrightarrow{L}}_0||{\overrightarrow{r}}_0||\overrightarrow{P}|$