Which of the following is//are conservative force (s) ?

Options

A. (a) `vecF=2r^3vecr`
B. (b) `vecF=-5/rhatr`
C. (c) `vecF=(3(xhati+yhatj))/((x^2+y^2)^(3//2))`
D. (d) `vecF=(3(yhati+xhatj))/((x^2+y^2)^(3//2)`

Explanation

To determine which of the given forces are conservative forces, we need to analyze each option based on the definition of conservative forces. A conservative force is one for which the work done in moving an object between two points is independent of the path taken. This means that the work done in a closed path is zero.

### Step-by-Step Solution:

1. **Understanding Conservative Forces**:
- A conservative force is defined by the property that the work done in moving an object in a closed loop is zero. This implies that the work done is path-independent.

2. **Analyzing Option A**:
- Given force: \( F = 2R^3 \hat{R} \)
- Work done \( W \) can be calculated using the integral:
\[ W = \int F \cdot dr \]
- Substituting \( F \):
\[ W = \int 2R^3 \, dR \]
- Integrating:
\[ W = \frac{2R^4}{4} + C = \frac{R^4}{2} + C \]
- Since we can find a potential function, option A is a conservative force.

3. **Analyzing Option B**:
- Given force: \( F = -\frac{5}{R} \hat{R} \)
- Work done:
\[ W = \int -\frac{5}{R} \, dR \]
- Integrating:
\[ W = -5 \ln R + C \]
- Since we can find a potential function, option B is also a conservative force.

4. **Analyzing Option C**:
- Given force: \( F = \frac{3x \hat{i} + 3y \hat{j}}{(x^2 + y^2)^{3/2}} \)
- Work done:
\[ W = \int \frac{3x \, dx + 3y \, dy}{(x^2 + y^2)^{3/2}} \]
- Using substitution \( t^2 = x^2 + y^2 \), we can simplify the integral.
- After simplification, we find that we can express the work done in a closed path, indicating that option C is a conservative force.

5. **Analyzing Option D**:
- Given force: \( F = y \hat{i} + x \hat{j} \)
- Work done:
\[ W = \int (y \, dx + x \, dy) \]
- This integral cannot be simplified to yield a potential function without additional information about the relationship between \( x \) and \( y \).
- Since we cannot express the work done in a closed path as zero, option D is not a conservative force.

### Conclusion:
- **Conservative Forces**: A, B, C
- **Non-Conservative Force**: D