Question

A small ball of mass $m$ suspended from the ceiling at a point O by a thread of length $\ell$ moves along a horizontal circle with a constant angular velocity $\omega$. Then,

• 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)

The correct options are
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

The angular moment about point C of the ball is the product of its linear momentum and radius, is given by
$\overrightarrow{L}=mvr$
Here, radius is constant hence tangential velocity is also constant and angular moment is conserved.
Also,
the angular moment about point O is
$\overrightarrow{L}=\overrightarrow{r}\times m\overrightarrow{v}$ ......(1).....( since, $\overrightarrow{L}$ is perpendicular to both $\overrightarrow{r}$ and $\overrightarrow{v}$)
Since both $\overrightarrow{r}$ and $\overrightarrow{v}$ are constant hence, vertical component of angular momentum about O is constant.
Now,
$\overrightarrow{L}=mrv\sin \theta$....... from(1)
here, $\theta ={90}^{\circ }$
$\overrightarrow{L}=mrv$
$\overrightarrow{L}=mr\left(r\omega \right)$
$\overrightarrow{L}=m{r}^2\omega =I\omega$
hence, magnitude of angular momentum about O is constant.