Question

The work done by the force $=A(y^2\hat{i}+2x^2\hat{j})$, where $A$ is a constant and $x$ and $y$ are in meters around the path shown is:

• A. ${\text{Ad}}^3$
• B. $\text{Ad}$
• C. ${\text{Ad}}^2$
• D. $\text{Zero}$

Answer 1

The correct option is A ${\text{Ad}}^3$

Find work done by the force in moving from $(0,0)$ to $(d,0)$.

Given,
force, $\overrightarrow{F}=A(y^2\hat{i}+2x^2\hat{j})$

Work done, $W=\int \overrightarrow{F}\cdot \overrightarrow{dx}$

$W=\int A(y^2\hat{i}+2x^2\hat{j})\cdot (dx\hat{i}+dy\hat{j})$

For the path $(0,0)$ to $(d,0)$, $y=0,dy=0$

$W_1=0+0=0$

Find work done by the force in moving from $(d,0)$ to $(d,d)$.

For the path $(d,0)$ to $(d,d)$,

$x=d\Rightarrow dx=0$

And
$dy=d$

$W_2=A[0+2d^{2}d]$

$W_2=2A{d}^3$

Find work done by the force in moving from $(d,d)$ to $(0,d)$.

For the path $(d,d)$ to $(0,d)$,

$y=d\Rightarrow dy=0$

And
$dx=-d$

$W_3=A\left[d^2\right(-d)+0]=-A{d}^3$

Find work done by the force in moving from $(0,d)$ to $(0,0)$.

For the path $(0,d)$ to $(0,0)$,

$x=d\Rightarrow dx=0$

And
$dy=-d$

$W_4=A[0+0]=0$

Total work done,
$W=W_1+W_2+W_3+W_4$
$W=0+2A{d}^3-A{d}^3+0W=A{d}^3$