A body is located at $(1,1,1) \mathrm{m}$ and experiences a force of $2 \mathrm{~N}$ in the direction…

A body is located at $(1,1,1) \mathrm{m}$ and experiences a force of $2 \mathrm{~N}$ in the direction $\hat{\mathrm{i}}+\hat{\mathrm{j}}$. The torque acting on the body in $\mathrm{N}-\mathrm{m}$ is
  1. $(-\sqrt{2} \hat{i}+\sqrt{2} \hat{j})$
  2. $(-\hat{i}+\hat{j})$
  3. $(\hat{i}-\hat{j})$
  4. $(\sqrt{2} \hat{i}+\sqrt{2} \hat{j})$

Solution

$\overrightarrow{\mathrm{F}}=2 \cdot \frac{1}{\sqrt{2}}(\hat{\mathrm{i}}+\hat{\mathrm{j}})=\sqrt{2}(\hat{\mathrm{i}}+\hat{\mathrm{j}})$ We have $\begin{aligned} & \overrightarrow{\mathrm{j}}=\overrightarrow{\mathrm{r}} \times \overrightarrow{\mathrm{F}}=\left|\begin{array}{ccc}\hat{\mathrm{i}} & \hat{\mathrm{j}} & \hat{\mathrm{k}} \\ 1 & 1 & 1 \\ \sqrt{2} & \sqrt{2} & 0\end{array}\right| \\ & =\hat{\mathrm{i}}(0-\sqrt{2})+\hat{\mathrm{j}}(\sqrt{2}-0)+\hat{\mathrm{k}}(0) \\ & =-\sqrt{2} \hat{\mathrm{i}}+\sqrt{2} \hat{\mathrm{j}}\end{aligned}$

Asked in: AP EAMCET 2022 (08 Jul Shift 1)

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