Which of the following has the same dimensions as those of angle ?

Options

A. `(E )/(M^(5)G)`
B. `(EJ^(2))/(M^(5)G^(2))`
C. `EM^(5)G^(-1)`
D. `EJ^(2)M^(5)G^(2)`

Explanation

To solve the question "Which of the following has the same dimensions as those of angle?", we need to determine the dimensional formula for angle and then compare it with the options provided.

### Step-by-Step Solution:

1. **Identify the Dimensional Formula for Angle:**
- The dimension of angle is dimensionless, represented as:
\[ [\text{Angle}] = M^0 L^0 T^0 \]
- This means that angle has no dimensions associated with mass (M), length (L), or time (T).

2. **Analyze the Options:**
- We need to evaluate each option provided in the question to see which one has the same dimensional formula as \( M^0 L^0 T^0 \).

3. **Option 1: \( \frac{E}{M^5 g} \)**
- The dimensional formula for energy (E) is:
\[ [E] = M^1 L^2 T^{-2} \]
- The dimensional formula for mass (M) is:
\[ [M] = M^1 L^0 T^0 \]
- The gravitational constant (g) has the dimensional formula:
\[ [g] = M^{-1} L^3 T^{-2} \]
- Now, substituting these into the expression:
\[ \text{Dimension of } \frac{E}{M^5 g} = \frac{M^1 L^2 T^{-2}}{M^5 \cdot M^{-1} L^3 T^{-2}} = \frac{M^1 L^2 T^{-2}}{M^{4} L^{3} T^{-2}} = M^{-3} L^{-1} T^{0} \]
- This does not match \( M^0 L^0 T^0 \).

4. **Option 2: \( \frac{E J^2}{M^5 g^2} \)**
- We already know the dimensional formula for energy (E):
\[ [E] = M^1 L^2 T^{-2} \]
- The angular momentum (J) has the dimensional formula:
\[ [J] = M^1 L^2 T^{-1} \]
- Therefore, \( J^2 \) will be:
\[ [J^2] = (M^1 L^2 T^{-1})^2 = M^2 L^4 T^{-2} \]
- The dimensional formula for \( g^2 \) is:
\[ [g^2] = (M^{-1} L^3 T^{-2})^2 = M^{-2} L^6 T^{-4} \]
- Now substituting these into the expression:
\[ \text{Dimension of } \frac{E J^2}{M^5 g^2} = \frac{M^1 L^2 T^{-2} \cdot M^2 L^4 T^{-2}}{M^5 \cdot M^{-2} L^6 T^{-4}} = \frac{M^{3} L^{6} T^{-4}}{M^{3} L^{6} T^{-4}} = M^0 L^0 T^0 \]
- This matches \( M^0 L^0 T^0 \).

5. **Conclusion:**
- The second option \( \frac{E J^2}{M^5 g^2} \) has the same dimensions as those of angle.

### Final Answer:
The correct option is **Option 2: \( \frac{E J^2}{M^5 g^2} \)**.