First slide

Rotational motion

Question

A force $\vec{F} = (\hat{i} + 2 \hat{j} + 3 \hat{k}) N$ acts at a point $(4 \hat{i} + 3 \hat{j} - \hat{k}) m$ . Then the magnitude of torque about the point $(\hat{i} + 2 \hat{j} + \hat{k}) m$ will be $\sqrt{x} N - m$ . The value of x is _____________

Easy

Solution

$\begin{array}{l} R e l a t i v e P o s i t i o n = \vec{r} = (4 - 1) \hat{i} + (3 - 2) \hat{j} + (- 1 - 1) \hat{k} \\ \Rightarrow \vec{r} = 3 \hat{i} + \hat{j} - 2 \hat{k} \\ A l s o, \vec{F} = \hat{i} + 2 \hat{j} + 3 \hat{j} \\ T o r q u e = \vec{τ} = \vec{r} \times \vec{F} = [\begin{matrix} \hat{i} & \hat{j} & \hat{k} \\ 3 & 1 & - 2 \\ 1 & 2 & 3 \end{matrix}] = 7 \hat{i} - 11 \hat{j} + 5 \hat{k} \\ M a g n i t u d e o f t o r q u e = \sqrt{7^{2} + 11^{2} + 5^{2}} = \sqrt{195} \end{array}$

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