Question

The work done by a force $\overrightarrow{F}=x^2\hat{i}+y^2\hat{j}$ around the path shown in the figure is:

• A. $\frac{2a^3}{3}$
• B. $0$
• C. $a^3$
• D. $\frac{4a^3}{3}$

Answer 1

Answer: (B)

$0$

The force acting $F(x,y)=x^2\hat{i}+y^2\hat{j}$

Thus we get $F_x=x^2$ and $F_y=y^2$

Now $\frac{\partial F_y}{\partial x}=$ $\frac{\partial y^2}{\partial x}=0$

Similarly, $\frac{\partial F_x}{\partial y}=$ $\frac{\partial x^2}{\partial y}=0$

For the force to be a conservative force, $\frac{\partial F_y}{\partial x}-\frac{\partial F_x}{\partial y}=0$

$⟹$ $F(x,y)$ is a conservative force.

Work done by a conservative force is zero for a closed loop.

$⟹$ Work done by force $F(x,y)$ around the path given is zero.