Which of the following forces in non conservation one ?

Options

A. `3hat(i) +4hat(j)`
B. `4xhat(i) + 3yhat(i)`
C. `3x^(2) hat(i) + 4y^(2) hat(j)`
D. `y^(2)hat(i) + x^(2)hat(j)`

Explanation

To determine which of the given forces is a non-conservative force, we first need to understand the definition of conservative and non-conservative forces.

### Step-by-Step Solution:

1. **Understanding Conservative Forces**:
- A conservative force is defined as a force for which the work done is independent of the path taken. The work done by conservative forces depends only on the initial and final positions.
- Examples of conservative forces include gravitational force and spring force.

2. **Identifying the Given Forces**:
- We have four options to evaluate:
- Option A: \(3 \hat{i} + 4 \hat{j}\)
- Option B: \(4x \hat{i} + 3y \hat{j}\)
- Option C: \(3x^2 \hat{i} + 4y^2 \hat{j}\)
- Option D: \(y^2 \hat{i} + x^2 \hat{j}\)

3. **Evaluating Each Option**:
- **Option A**: \(3 \hat{i} + 4 \hat{j}\)
- This is a constant force and does not depend on position. It is a conservative force.
- **Option B**: \(4x \hat{i} + 3y \hat{j}\)
- This force depends on the position (x and y). It meets the criteria for a conservative force.
- **Option C**: \(3x^2 \hat{i} + 4y^2 \hat{j}\)
- This force also depends on the position (x and y). It is a conservative force.
- **Option D**: \(y^2 \hat{i} + x^2 \hat{j}\)
- Here, the force components do not follow the required format of having \(x\) with \(\hat{i}\) and \(y\) with \(\hat{j}\). This indicates that the work done by this force depends on the path taken, making it a non-conservative force.

4. **Conclusion**:
- Based on the evaluations, the non-conservative force among the options is **Option D: \(y^2 \hat{i} + x^2 \hat{j}\)**.

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

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