Question

A groove is in the form of a broken line ABC and the position vectors of the three points are respectively $2\mathbf{i} - 3\mathbf{j} + 2\mathbf{k}$, $3\mathbf{i} + 2\mathbf{j} - \mathbf{k}$ and $\mathbf{i} + \mathbf{j} + \mathbf{k}$. A force of magnitude $24\sqrt{3}$ acts on a particle of unit mass kept at the point A and moves it along the groove to the point C. If the line of action of the force is parallel to the vector $\mathbf{i} + 2\mathbf{j} + \mathbf{k}$ all along, the number of units of work done by the force is

• A. $144\sqrt{2}$
• B. $144\sqrt{3}$
• C. $72\sqrt{2}$
• D. $72\sqrt{3}$

Answer 1

Answer: (C)

$72\sqrt{2}$

Answer 2

$\overset{\rightarrow}{F} = (24\sqrt{3})$ $\frac{\mathbf{i} + 2\mathbf{j} + \mathbf{k}}{|\mathbf{i} + 2\mathbf{j} + \mathbf{k}|}$= $\frac{24\sqrt{3}}{\sqrt{6}}(\mathbf{i} + 2\mathbf{j} + \mathbf{k})$

= $12\sqrt{2}(\mathbf{i} + 2\mathbf{j} + \mathbf{k})$

Displacement $\mathbf{r} =$ position vector of C – Position vector of A = $(\mathbf{i} + \mathbf{j} + \mathbf{k}) - (2\mathbf{i} - 3\mathbf{j} + 2\mathbf{k})$ = $( - \mathbf{i} + 4\mathbf{j} - \mathbf{k})$

Work done by the force $W = \mathbf{r}.\overset{\rightarrow}{F}$

= $( - \mathbf{i} + 4\mathbf{j} - \mathbf{k}).12\sqrt{2}(\mathbf{i} + 2\mathbf{j} + \mathbf{k})$ = $12\sqrt{2}( - 1 + 8 - 1) = 72\sqrt{2}$