Which of the following force is not conservative :-

Options

A. `bar(F)=3hati+4hatj`
B. `bar(F)=3xhati+4yhatj`
C. `bar(F)=3yhati+4xhatj`
D. `bar(F)=3x^(2)hati+4y^(2)hatj`

Answer

To determine which of the given forces is not conservative, we need to understand the definition of conservative forces. A conservative force is one where the work done by the force on an object moving between two points is independent of the path taken. This means that the work done only depends on the initial and final positions of the object.

### Step-by-step Solution:

1. **Understand Conservative Forces**:
- A conservative force is defined as one where the work done is path-independent. Mathematically, if a force \( \vec{F} \) can be expressed as \( \vec{F} = P(x, y) \hat{i} + Q(x, y) \hat{j} \), then it is conservative if the following condition holds:
\[ \frac{\partial Q}{\partial x} = \frac{\partial P}{\partial y} \]

2. **Analyze Each Option**:
- **Option A**: \( \vec{F} = 3 \hat{i} + 4 \hat{j} \)
- Here, \( P = 3 \) and \( Q = 4 \).
- Since both \( P \) and \( Q \) are constants, the derivatives \( \frac{\partial Q}{\partial x} = 0 \) and \( \frac{\partial P}{\partial y} = 0 \) are equal. Thus, this force is conservative.

- **Option B**: \( \vec{F} = 3x \hat{i} + 4y \hat{j} \)
- Here, \( P = 3x \) and \( Q = 4y \).
- Calculate the derivatives:
\[ \frac{\partial Q}{\partial x} = 0 \quad \text{and} \quad \frac{\partial P}{\partial y} = 0 \]
- Since both derivatives are equal, this force is also conservative.

- **Option C**: \( \vec{F} = 3y \hat{i} + 4x \hat{j} \)
- Here, \( P = 3y \) and \( Q = 4x \).
- Calculate the derivatives:
\[ \frac{\partial Q}{\partial x} = 4 \quad \text{and} \quad \frac{\partial P}{\partial y} = 3 \]
- Since \( 4 \neq 3 \), this force is not conservative.

- **Option D**: \( \vec{F} = 3x^2 \hat{i} + 4y^2 \hat{j} \)
- Here, \( P = 3x^2 \) and \( Q = 4y^2 \).
- Calculate the derivatives:
\[ \frac{\partial Q}{\partial x} = 0 \quad \text{and} \quad \frac{\partial P}{\partial y} = 0 \]
- Since both derivatives are equal, this force is conservative.

3. **Conclusion**:
- The only non-conservative force among the options is **Option C**: \( \vec{F} = 3y \hat{i} + 4x \hat{j} \).

### Final Answer:
**Option C** is the non-conservative force.