Question

A force $\overrightarrow{F}=(2\hat{i}+3\hat{j}-5\hat{k})N$ acts at a point ${\overrightarrow{r}}_1=(2\hat{i}+4\hat{j}+7\hat{k})m$. The torque of the force about the point ${\overrightarrow{r}}_2=(\hat{i}+2\hat{j}+3\hat{k})m$ is :

• A. $17\hat{j}+5\hat{k}-3\hat{i})Nm$
• B. $2\hat{i}+4\hat{j}-6\hat{k})Nm$
• C. $12\hat{i}-5\hat{k}+7\hat{k})Nm$
• D. $13\hat{j}-22\hat{i}-\hat{k})Nm$

Answer 1

The correct option is D $13\hat{j}-22\hat{i}-\hat{k})Nm$

Consider the problem

Force $\overrightarrow{F}=(2\hat{i}+3\hat{j}-5\hat{k})$ is acted at a point ${\overrightarrow{r}}_1=(2\hat{i}+4\hat{j}+7\hat{k})$

Torque about the point $0,{\overrightarrow{r}}_2=(\hat{i}+2\hat{j}+3\hat{k})$ is

$\overrightarrow{r}\times \overrightarrow{F}=({\overrightarrow{r}}_1-{\overrightarrow{r}}_2)\times \overrightarrow{F}$

And

$({\overrightarrow{r}}_1-{\overrightarrow{r}}_2)=\hat{i}+2\hat{j}+4\hat{k}$

$\overrightarrow{r}\times \overrightarrow{F}=\left|\begin{array}{ccc}\hat{i}& \hat{j}& \hat{k}\\ 1& 2& 4\\ 2& 3& -5\end{array}\right|$

$=\left(-10-12\right)\hat{i}+\left(8-\left(-5\right)\right)\hat{j}+\left(3-4\right)\hat{k}$

$=-22\hat{i}+13\hat{j}-\hat{k}$

Therefore, the torque of the force about the point

$=13\hat{j}-22\hat{i}-\hat{k})Nm$

Hence, Option $D$ is the correct answer.