Question

A particle acted upon by constant forces $F_1=4\hat{i}+\hat{j}-3\hat{k}$ and $F_2=3\hat{i}+\hat{j}-\hat{k}$ is displaced from the point $r_1=\hat{i}+2\hat{j}-3\hat{k}$ to point $r_2=5\hat{i}+4\hat{j}-\hat{k}$. The total work done by the forces in SI unit is

• A. $20$
• B. $24$
• C. $50$
• D. $30$
• E. $35$

Answer 1

The correct option is B

$24$

Step 1: Given data.

Force, $F_1=4\hat{i}+\hat{j}-3\hat{k}$

Force, $F_2=3\hat{i}+\hat{j}-\hat{k}$

Displacement, $r_1=\hat{i}+2\hat{j}-3\hat{k}$

Displacement, $r_2=5\hat{i}+4\hat{j}-\hat{k}$

Step 2: Find the total work done by the forces.

The net force is,

$$\begin{gathered} F=F_1+F_2 \\ F=4\hat{i}+\hat{j}-3\hat{k}+3\hat{i}+\hat{j}-\hat{k} \\ F=7\hat{i}+2\hat{j}-4\hat{k} \end{gathered}$$

Net Displacement is,

$$\begin{gathered} d=r_2-r_1 \\ d=(5\hat{i}+4\hat{j}-\hat{k})-(\hat{i}+2\hat{j}-3\hat{k}) \\ d=4\hat{i}+2\hat{j}+2\hat{k} \end{gathered}$$

The work done is given by,

$$\begin{gathered} W=F.d \\ W=(7\hat{i}+2\hat{j}-4\hat{k}).(4\hat{i}+2\hat{j}+2\hat{k}) \\ W=28+4-8 \\ W=24 \end{gathered}$$

Hence, option B is correct.