Question

A constant force $\overrightarrow{F}=2\hat{i}-3\hat{j}+\hat{k}$ is applied on a ball to displace it from $\overrightarrow{r_1}=\hat{i}+\hat{j}$ to $\overrightarrow{r_2}=4\hat{i}+3\hat{j}$, which of the following statements is true?

• A. Force and displacement are perpendicular to each other and work done by force is equal to zero
• B. Force and displacement are parallel to each other and work done by force is equal to zero
• C. Force and displacement are anti parallel to each other and work done by force is equal to zero
• D. Force and displacement are perpendicular to each other and work done by force is not equal to zero

Answer 1

The correct option is A Force and displacement are perpendicular to each other and work done by force is equal to zero

Given constant force $\overrightarrow{F}=(2\hat{i}-3\hat{j}+\hat{k})\text{N}$
Initial position $\overrightarrow{r_1}=(\hat{i}+\hat{j})\text{m}$
Final position $\overrightarrow{r_2}=(4\hat{i}+3\hat{j})\text{m}$

To find the displacement $\left(\overrightarrow{s}\right)=\overrightarrow{r_2}-\overrightarrow{r_1}$
$=(4\hat{i}+3\hat{j})-(\hat{i}+\hat{j})=(3\hat{i}+2\hat{j})\text{m}$

To find angle between $\overrightarrow{F}$ and $\overrightarrow{s}$:
We know, $\overline{A}.\overline{B}=AB\cos \theta$
$\Rightarrow \cos \theta =\frac{\overline{A}.\overline{B}}{AB}$
Similarly, $\cos \theta =\frac{\overrightarrow{F}.\overrightarrow{s}}{\left|F\right|\left|s\right|}$
$=\frac{(2\hat{i}-3\hat{j}+\hat{k}).(3\hat{i}+2\hat{j})}{\sqrt{4^2+3^2+1^2}\sqrt{3^2+2^2}}=\frac{6-6}{\sqrt{26}\sqrt{13}}=0$
Since, $\cos \theta =0\Rightarrow \theta ={90}^{\text{o}}$
$\Rightarrow \overrightarrow{F}\perp \overrightarrow{s}$
Since force and displacement are perpendicular to each other the work done is equal to zero.

Hence option A is the correct answer.