Question

A force $\overrightarrow{F}=\alpha \hat{i}+3\hat{j}+6\hat{k}$ is acting at a point $\overrightarrow{r}=2\hat{i}-6\hat{j}-12\hat{k}$ . The value of $\alpha$ for which angular momentum about origin is conversed is ?

• A. 1
• B. -1
• C. 2
• D. Zero

Answer 1

The correct option is B -1

For the conservation of angular momentum about origin, the torque $t$ acting on the particle will be zero.

$\begin{array}{c}\overrightarrow{\tau }=\overrightarrow{r}\times \overrightarrow{F}\\ \overrightarrow{r}=2\hat{i}-6\hat{j}-12\overrightarrow{k}\\ and\overrightarrow{F}=a\hat{i}+3\hat{j}+6\overrightarrow{k}\\ \therefore \overrightarrow{\tau }=\left|\begin{array}{ccc}\overrightarrow{i}& \overrightarrow{j}& \overrightarrow{k}\\ 2& -6& -12\\ \alpha & 3& 6\end{array}\right|\\ =\overrightarrow{i}\left(-36+36\right)-\overrightarrow{j}\left(12+12\alpha \right)+\overrightarrow{k}\left(6+6\alpha \right)\\ =-\overrightarrow{j}\left(12+12\alpha \right)+\overrightarrow{k}\left(6+6\alpha \right)\\ But\overrightarrow{\tau }=0\\ \therefore 12+12\alpha =0\alpha =-1\\ and6+6\alpha =0or\alpha =-1\end{array}$

Hence, Option $B$ is correct.