Question

A particle is acted upon by constant forces $4\overline{i}+\overline{j}-3\overline{k}$ and $3\overline{i}+\overline{j}-\overline{k}$ which displace it from a point $\overline{i}+2\overline{j}+3\overline{k}$ to the point $5\overline{i}+4\overline{j}+\overline{k}$ . The work done in standard unity by the forces is given by

• A. $30$
• B. $50$
• C. $40$
• D. $60$

Answer 1

The correct option is C $40$

$\begin{array}{cc}\overrightarrow{F_1}=4\hat{i}+\hat{j}-3\hat{k}\&\overrightarrow{F_2}=3\hat{i}+\hat{j}-\hat{k}& \\ F=\overrightarrow{F_1}+\overrightarrow{F_2}\Rightarrow \left(4\hat{i}+\hat{j}-3\hat{k}\right)+\left(3\hat{i}+\hat{j}-\hat{k}\right)=7\hat{i}+2\hat{j}-4\hat{k}& \\ \overrightarrow{d}=\overrightarrow{d_2}-\overrightarrow{d_1}\Rightarrow \left(5\hat{i}+4\hat{j}+\hat{k}\right)-\left(\hat{i}+2\hat{j}+3\hat{k}\right)& \\ =\left(5\hat{i}+4\hat{j}+\hat{k}\right)-\hat{i}-2\hat{j}-3\hat{k}=4\hat{i}+2\hat{j}-2\hat{k}& \\ w=\overrightarrow{f}.\overrightarrow{d}& \\ w=\left(7\hat{i}+2\hat{j}-4\hat{k}\right)\left(4\hat{i}+2\hat{j}-2\hat{k}\right)& \\ w=28+4+8& \left(\hat{i}.\hat{i}=\hat{j}.\hat{j}=\hat{k}.\hat{k}=1\right)\\ w=40& \end{array}$