Question

Choose the correct option for the two given statements from the options:

Statement-$1$: The torque about the point $2\hat{i}+\hat{j}-\hat{k}$ of a force represented by $4\hat{i}+\hat{k}$ acting through the point $\hat{i}-\hat{j}+2\hat{k}$ is $-2\hat{i}+13\hat{j}+8\hat{k}$.

Statement-$2$: The torque of a force $\overrightarrow{F}$ about a point $P$ is $\overrightarrow{r}\times \overrightarrow{F}$ where $\overrightarrow{r}$ is the vector from the point $P$ to any point $\overrightarrow{a}$ on the line of action $L$ of $\overrightarrow{F}$.

• A. Statement-$1$ is true, statement-$2$ is true, statement-$2$ is a correct explanation for statement-$1$.
• B. Statement-$1$ is true, statement-$2$ is true, statement-$2$ is not a correct explanation for statement-$1$.
• C. Statement-$1$ is true, statement-$2$ is false
• D. Statement-$1$ is false, statement-$2$ is true

Answer 1

The correct option is A Statement-$1$ is true, statement-$2$ is true, statement-$2$ is a correct explanation for statement-$1$.

$\overline{F}=4\overline{i}+\overline{k}$
Force acting on $\overline{i}-\overline{j}+2\overline{k}$ and torque taken about the point $2\overline{i}+\overline{j}-\overline{k}.$
Thus, $\overline{r}=-\overline{i}-2\overline{j}+3\overline{k}$
$\overline{P}=\overline{r}\times \overline{F}=\overline{j}+8\overline{k}-2\overline{i}+12\overline{j}$
$=-2\overline{i}+13\overline{j}+8\overline{k}$

The statement II is correct by definition as the torque of a force $\overrightarrow{F}$ about a point $P$ is $\overrightarrow{r}\times \overrightarrow{F}$ where $\overrightarrow{r}$ is the vector from the point $P$ to any point $\overrightarrow{a}$ on the line of action $L$ of $\overrightarrow{F}$.

Thus statement II is the correct explanation for statement I.