The potential energy function of a particle in the x-y plane is given by `U =k(x+y)`, where (k) is a constant. The work done by the conservative force in moving a particlae from (1,1) to (2,3) is .

Options

A. `-3 k`
B. `+3 k`
C. `k`
D. None of these

Explanation

To solve the problem, we need to calculate the work done by the conservative force when moving a particle from the point (1, 1) to (2, 3) using the given potential energy function \( U = k(x + y) \).

### Step-by-Step Solution:

1. **Identify the Initial and Final Points**:
- Initial point \( (x_1, y_1) = (1, 1) \)
- Final point \( (x_2, y_2) = (2, 3) \)

2. **Calculate the Initial Potential Energy \( U_i \)**:
- Using the potential energy function \( U = k(x + y) \):
\[ U_i = k(1 + 1) = k \cdot 2 = 2k \]

3. **Calculate the Final Potential Energy \( U_f \)**:
- For the final point \( (2, 3) \):
\[ U_f = k(2 + 3) = k \cdot 5 = 5k \]

4. **Calculate the Change in Potential Energy \( \Delta U \)**:
- The change in potential energy is given by:
\[ \Delta U = U_f - U_i = 5k - 2k = 3k \]

5. **Calculate the Work Done by the Conservative Force**:
- The work done by the conservative force is the negative of the change in potential energy:
\[ W = -\Delta U = -3k \]

### Final Answer:
The work done by the conservative force in moving the particle from (1, 1) to (2, 3) is \( -3k \).

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