Question

The time dependence of the position of a particle of mass $m$ $=2$ is given by $\overrightarrow{r}\left(t\right)=2t\hat{i}-3t^2\hat{j}.$ Its angular momentum, with respect to the origin, at time $t=2$ is :

• A. $36\hat{k}$
• B. $-34(\hat{k}-\hat{i})$
• C. $-48\hat{k}$
• D. $48(\hat{i}\times \hat{j})$

Answer 1

The correct option is C $-48\hat{k}$

Given:
mass $m=2$
time $t=2$
$\overrightarrow{r}=2t\hat{i}-3t^2\hat{j}$
Now, Position at time $t=2$
$\overrightarrow{r}\text{(at t=2)}=(2\times 2)\hat{i}-(3\times 2^2)\hat{j}=4\hat{i}-12\hat{j}$
Velocity,
$\overrightarrow{v}=\frac{d\overrightarrow{r}}{dt}=\frac{d}{dt}(2t\hat{i}-3t^2\hat{j})=2\hat{i}-6t\hat{j}$

Velocity at time $t=2$

$\overrightarrow{v}\left(\text{at t=2}\right)=2\hat{i}-(6\times 2)\hat{j}=2\hat{i}-12\hat{j}$

Now, angular momentum
$\overrightarrow{L}=mvr\sin \theta \hat{n}=m(\overrightarrow{r}\times \overrightarrow{v})$

$=2(4\hat{i}-12\hat{j})\times (2\hat{i}-12\hat{j})=-48\hat{k}$

Hence option $\left(D\right)$ is correct.