Question

Three forces $\mathbf{i} + 2\mathbf{j} - 3\mathbf{k},2\mathbf{i} + 3\mathbf{j} + 4\mathbf{k}$ and $\mathbf{i} - \mathbf{j} + \mathbf{k}$ are acting on a particle at the point (0, 1, 2). The magnitude of the moment of the forces about the point (1, – 2, 0) is

• A. $\frac{1}{3}(\mathbf{p} + \mathbf{q} + \mathbf{r})$
• B. $6\sqrt{10}$
• C. $4\sqrt{17}$
• D. None of these

Answer 1

Answer: (B)

$6\sqrt{10}$

Answer 2

Total force

$\overset{\rightarrow}{F} = (\mathbf{i} + 2\mathbf{j} - 3\mathbf{k}) + (2\mathbf{i} + 3\mathbf{j} + 4\mathbf{k}) + (\mathbf{i} - \mathbf{j} + \mathbf{k}) = 4\mathbf{i} + 4\mathbf{j} + 2\mathbf{k}$Moment of the forces about P = $\mathbf{r} \times \overset{\rightarrow}{F} = \overset{\rightarrow}{PA} \times \overset{\rightarrow}{F}$

$\overset{\rightarrow}{PA} = (0 - 1)\mathbf{i} + (1 + 2)\mathbf{j} + (2 - 0)\mathbf{k} = - \mathbf{i} + 3\mathbf{j} + 2\mathbf{k}$

$\therefore$ Moment about P = $( - \mathbf{i} + 3\mathbf{j} + 2\mathbf{k}) \times (4\mathbf{i} + 4\mathbf{j} + 2\mathbf{k})$

=$\left| \begin{matrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \

• 1 & 3 & 2 \ 4 & 4 & 2 \end{matrix} \right| = - 2\mathbf{i} + 10\mathbf{j} - 16\mathbf{k}$

Magnitude of the moment = $| - 2\mathbf{i} + 10\mathbf{j} - 16\mathbf{k}|$

= $2\sqrt{1^{2} + 5^{2} + 8^{2}} = 2\sqrt{90} = 6\sqrt{10}$