Question

A force $\overrightarrow{F}=(\hat{i}+2\hat{j})\text{N}$ acts at $O$, the origin of the co-ordinate system. The torque about the point $(1,–1,1)\text{m}$ is

• A. $(2\hat{i}-\hat{j}-3\hat{k})\text{Nm}$
• B. $(\hat{i}-\hat{j}-\hat{k})\text{Nm}$
• C. $(2\hat{i}+2\hat{j}-\hat{k})\text{Nm}$
• D. $(-\hat{i}+\hat{j}+2\hat{k})\text{Nm}$

Answer 1

The correct option is A $(2\hat{i}-\hat{j}-3\hat{k})\text{Nm}$

Torque of force $\overrightarrow{F}$ acting at $\overrightarrow{r}$ is given by
$\overrightarrow{\tau }=\overrightarrow{r}\times \overrightarrow{F}$
Here $\overrightarrow{r}=(0-1)\hat{i}+(0+1)\hat{j}+(0-1)\hat{k}$
$\overrightarrow{r}=-\hat{i}+\hat{j}-\hat{k}$

$\overrightarrow{\tau }=\left|\begin{array}{ccc}\hat{i}& \hat{j}& \hat{k}\\ -1& 1& -1\\ 1& 2& 0\end{array}\right|$
$\overrightarrow{\tau }=[0-(-2\left)\right]\hat{i}-\hat{j}[0-(-1\left)\right]+\hat{k}(-2-1)$
$\overrightarrow{\tau }=(2\hat{i}-\hat{j}-3\hat{k})\text{Nm}$