If `E , M , J , and G` , respectively , denote energy , mass , angular momentum , and gravitational constant , then `EJ^(2) //M^(5) G^(2)` has the dimensions of

Options

A. mass
B. length
C. time
D. angle

Explanation

To find the dimensions of the expression \( \frac{E J^2}{M^5 G^2} \), we will break down the dimensions of each variable involved: energy (E), angular momentum (J), mass (M), and gravitational constant (G).

### Step 1: Determine the dimensions of each variable.
1. **Energy (E)**:
- The dimension of energy is given by:
\[ [E] = M^1 L^2 T^{-2} \]

2. **Angular Momentum (J)**:
- The dimension of angular momentum is given by:
\[ [J] = M^1 L^2 T^{-1} \]

3. **Mass (M)**:
- The dimension of mass is:
\[ [M] = M^1 \]

4. **Gravitational Constant (G)**:
- The dimension of the gravitational constant is:
\[ [G] = M^{-1} L^3 T^{-2} \]

### Step 2: Substitute the dimensions into the expression.
Now we substitute these dimensions into the expression \( \frac{E J^2}{M^5 G^2} \).

- Substitute for \( E \):
\[ E = M^1 L^2 T^{-2} \]

- Substitute for \( J^2 \):
\[ J^2 = (M^1 L^2 T^{-1})^2 = M^2 L^4 T^{-2} \]

- Substitute for \( M^5 \):
\[ M^5 = (M^1)^5 = M^5 \]

- Substitute for \( G^2 \):
\[ G^2 = (M^{-1} L^3 T^{-2})^2 = M^{-2} L^6 T^{-4} \]

### Step 3: Combine the dimensions.
Now we can combine these dimensions in the expression:
\[ \frac{E J^2}{M^5 G^2} = \frac{(M^1 L^2 T^{-2})(M^2 L^4 T^{-2})}{M^5 (M^{-2} L^6 T^{-4})} \]

### Step 4: Simplify the expression.
Let's simplify the numerator and the denominator separately.

- **Numerator**:
\[ E J^2 = M^1 L^2 T^{-2} \cdot M^2 L^4 T^{-2} = M^{1+2} L^{2+4} T^{-2-2} = M^3 L^6 T^{-4} \]

- **Denominator**:
\[ M^5 G^2 = M^5 \cdot M^{-2} L^6 T^{-4} = M^{5-2} L^6 T^{-4} = M^3 L^6 T^{-4} \]

Now, substituting back into the expression:
\[ \frac{M^3 L^6 T^{-4}}{M^3 L^6 T^{-4}} = M^{3-3} L^{6-6} T^{-4+4} = M^0 L^0 T^0 \]

### Step 5: Conclusion.
Since \( M^0 L^0 T^0 \) indicates that the expression is dimensionless, we can conclude that:
\[ \text{The dimensions of } \frac{E J^2}{M^5 G^2} \text{ is } M^0 L^0 T^0 \]
This corresponds to the dimensions of an angle, which is also dimensionless.

### Final Answer:
The dimensions of \( \frac{E J^2}{M^5 G^2} \) is \( M^0 L^0 T^0 \) (dimensionless).

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