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. Time
B. Angle
C. Mass
D. Length

Explanation

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

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

2. **Mass (M)**: The dimension of mass is simply:
\[ [M] = M \]

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

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 can substitute these dimensions into the expression \( \frac{E J^2}{M^5 G^2} \).

1. Substitute the dimensions:
\[ [E J^2] = [E] \cdot [J]^2 = (M L^2 T^{-2}) \cdot (M L^2 T^{-1})^2 \]

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

3. Now substitute back:
\[ [E J^2] = (M L^2 T^{-2}) \cdot (M^2 L^4 T^{-2}) = M^{1+2} L^{2+4} T^{-2-2} = M^3 L^6 T^{-4} \]

### Step 3: Calculate the dimensions of the denominator
1. For \( M^5 \):
\[ [M^5] = M^5 \]

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

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

### Step 5: Simplify the expression
Upon simplification, we find:
\[ \frac{E J^2}{M^5 G^2} = 1 \]

### Conclusion
The dimensions of the expression \( \frac{E J^2}{M^5 G^2} \) is dimensionless.