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 find the work done by the conservative force in moving a particle from point (1,1) to point (2,3) given the potential energy function \( U = k(x+y) \), we can follow these steps:

### Step 1: Identify the potential energy function
The potential energy function is given as:
\[ U = k(x + y) \]
where \( k \) is a constant.

### Step 2: Determine the initial and final positions
The initial position is \( (x_1, y_1) = (1, 1) \) and the final position is \( (x_2, y_2) = (2, 3) \).

### Step 3: Calculate the initial potential energy
Substituting the initial coordinates into the potential energy function:
\[ U_{\text{initial}} = U(1, 1) = k(1 + 1) = k \cdot 2 = 2k \]

### Step 4: Calculate the final potential energy
Substituting the final coordinates into the potential energy function:
\[ U_{\text{final}} = U(2, 3) = k(2 + 3) = k \cdot 5 = 5k \]

### Step 5: Calculate the change in potential energy
The change in potential energy \( \Delta U \) is given by:
\[ \Delta U = U_{\text{final}} - U_{\text{initial}} = 5k - 2k = 3k \]

### Step 6: Calculate the work done by the conservative force
The work done by the conservative force \( W_c \) is equal to the negative change in potential energy:
\[ W_c = -\Delta U = -3k \]

### Final Answer
Thus, the work done by the conservative force in moving the particle from (1,1) to (2,3) is:
\[ W_c = -3k \]

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