Question

A force $\overrightarrow{F}=(\hat{i}–\hat{j}+\hat{k})\text{N}$ acts at point $P$. Find the torque due to the force $\overrightarrow{F}$ about $O$.

• A. $\hat{i}+6\hat{j}+5\hat{k}\text{Nm}$
• B. $5\hat{i}-6\hat{j}+\hat{k}\text{Nm}$
• C. $5\hat{i}+6\hat{j}+\hat{k}\text{Nm}$
• D. $-\hat{i}-6\hat{j}-5\hat{k}\text{Nm}$
  

Answer 1

The correct option is D $-\hat{i}-6\hat{j}-5\hat{k}\text{Nm}$


Given, Force $\overrightarrow{F}=(\hat{i}–\hat{j}+\hat{k})\text{N}$
${\overrightarrow{r}}_1=(-\hat{i}+\hat{j}-2\hat{k})\text{m}$
${\overrightarrow{r}}_2=(2\hat{i}+3\hat{j}-5\hat{k})\text{m}$
$OP=\overrightarrow{r}={\overrightarrow{r}}_2-{\overrightarrow{r}}_1$
$\Rightarrow \overrightarrow{r}=(2\hat{i}+3\hat{j}-5\hat{k})-(-\hat{i}+\hat{j}-2\hat{k})$
$\therefore \overrightarrow{r}=(3\hat{i}+2\hat{j}-3\hat{k})\text{m}$
Torque $\overrightarrow{\tau }$ about $O$ is given by,
$\overrightarrow{\tau }=\overrightarrow{r}\times \overrightarrow{F}$
Now, $\overrightarrow{r}\times \overrightarrow{F}=\left|\begin{array}{ccc}\hat{i}& \hat{j}& \hat{k}\\ 3& 2& -3\\ 1& -1& 1\end{array}\right|=-\hat{i}-6\hat{j}-5\hat{k}$
$\therefore \overrightarrow{\tau }=-\hat{i}-6\hat{j}-5\hat{k}\text{Nm}$