Question

If force $\overrightarrow{F}=y\hat{i}+x\hat{j}$ acting on a particle then find the work done in moving the particle from position $(2,3)$ to $(5,6)$

• A. <!--td {border: 1px solid #ccc;}br {mso-data-placement:same-cell;}--> $18\text{J}$
• B. $24\text{J}$
• C. $30\text{J}$
• D. $36\text{J}$

Answer 1

The correct option is B $24\text{J}$

We know that, $$dW=\overrightarrow{F}\cdot \overrightarrow{ds}$$Given, $\overrightarrow{F}=y\hat{i}+x\hat{j}$ and take $\overrightarrow{ds}=(dx\hat{i}+dy\hat{j})$

$\Rightarrow dW=(y\hat{i}+x\hat{j})\cdot (dx\hat{i}+dy\hat{j})$

$\Rightarrow dW=ydx+xdy$

Now,
$ydx+xdy=d\left(xy\right)\text{(perfect differential equation)}$

$\Rightarrow dW=d\left(xy\right)$

For total work done, integrate both side with proper limits,

${\int }_0^{W}dW={\int }_{k_i}^{k_f}d\left(xy\right)$

$\text{Put}xy=k$

Then at $(2,3)\to k_i=2\times 3=6(5,6)\to k_f=5\times 6=30$

$\therefore W={\int }_6^{30}dk=[k{]}_6^{30}$

$\Rightarrow W=(30-6)=24\text{J}$

Note: In the above case work done depends only on initial and final positions and is independent of the path followed.
Hence, the force here is conservative force.

Thus, option (b) is the correct answer.

Key concept : Conservative force A force is said to be conservative if works done by or against the force in moving a body depends only on the initial and final positions of the body, and does not depend on the nature of the path followed between the initial and final positions. Examples of conservative forces are : gravitational force, elastic spring force and electrostatic force.