A particle is moving in xy plane; let F = (y i  x j) / (x^2 + y^2) be one of the forces acting on it. All coordinates are measured in S.I units. Which of the following option is/... A particle is moving in xy plane; let F = (y i  x j) / (x^2 + y^2) be one of the forces acting on it. All coordinates are measured in S.I units. Which of the following option is/are incorrect? A) Work done by this F when the particle moves in the triangular loop ABCA is zero. B) Work done by this force in the open path MNQ is π/2. C) Work done by this force when the particle moves on the circle of radius 1 m centered at (5,0) is zero. D) F is conservative.
Understand the Problem
The question is asking about the work done by a force acting on a particle moving in the xy plane, specifically which of the provided options regarding the work done under different paths and conditions is incorrect.
Answer
B, D
Answer for screen readers
The incorrect options are B) work done in the open path MNQ is $\frac{\pi}{2}$ and D) $\mathbf{F}$ is conservative.
Steps to Solve

Identify the Force Field Type Determine if the force field $ \mathbf{F} = \frac{y \mathbf{i}  x \mathbf{j}}{x^2 + y^2} $ is conservative. A force is conservative if the work done in a closed path is zero.

Check Work Done Along Closed Path (ABCA) Evaluate if the work done around the triangular loop ABCA is zero. Using the line integral, if the path is closed, calculate: $$ W_{closed} = \oint_C \mathbf{F} \cdot d\mathbf{r} $$ If it is a closed loop and the field is conservative, the result should be zero.

Evaluate Work Done in Open Path (MNQ) Calculate the work done along the open path from M to N to Q. The work can be calculated using: $$ W_{MNQ} = \int_{M}^{N} \mathbf{F} \cdot d\mathbf{r} + \int_{N}^{Q} \mathbf{F} \cdot d\mathbf{r} $$ Check if this integral equals $ \frac{\pi}{2} $.

Evaluate Work Along Circle Centered at (5,0) Calculate the work done when the particle moves along the circle of radius 1 m centered at (5,0). This path should also be evaluated using the line integral to see if $ W = 0 $.

Determine Conservativeness of the Force Field To check if the force is conservative, compute the curl of the force field: $$ \nabla \times \mathbf{F} = 0 $$ If the curl is not zero at any point, then the field is not conservative.
The incorrect options are B) work done in the open path MNQ is $\frac{\pi}{2}$ and D) $\mathbf{F}$ is conservative.
More Information
The given force field is not conservative due to a singularity at the origin, which affects the calculation of work. Option B indicates specific calculated work that may not hold with the correct method.
Tips
 Confusing conservative force with nonconservative due to overlooking singularities.
 Miscalculating the work done along paths due to incorrect parameterization or line integrals.