A force $ \vec{F}=5 \hat{i}+2 \hat{j}-5 \hat{k} $ acts on a particle whose position vector is $ \vec{r}=\hat{i}-2 \hat{j}+\hat{k} $. What is the torque about the origin ?

Question

A force $ \vec{F}=5 \hat{i}+2 \hat{j}-5 \hat{k} $ acts on a particle whose position vector is $ \vec{r}=\hat{i}-2 \hat{j}+\hat{k} $. What is the torque about the origin ?, $ 8 \hat{i}+10 \hat{j}+12 \hat{k} $, $ 8 \hat{i}+10 \hat{j}-12 \hat{k} $, $ 8 \hat{i}-10 \hat{j}-8 \hat{k} $, $ 10 \hat{i}-10 \hat{j}-\hat{k} $

MARKS App · Accepted Answer

Given $ \vec{F}=5 \hat{i}+2 \hat{j}-5 \hat{k} ; \vec{r}=\hat{i}-2 \hat{j}+\hat{k} $ We know torque $$ \tau=\vec{r} \times \vec{F}=\left|\begin{array}{ccc} \hat{i} & \hat{j} & \hat{k} \ 1 & -2 & 1 \ 5 & 2 & -5 \end{array}\right| $$ Therefore, $$ \begin{array}{l} \tau=\hat{i}(10-2)-\hat{j}(-5-5)+\hat{k}(2+10) \ \Rightarrow \tau=8 \hat{i}+10 \hat{j}+12 \hat{k} \end{array} $$ Thus, torque about origin is $ 8 \hat{i}+10 \hat{j}+12 \hat{k} $

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