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. angle
B. length
C. mass
D. time

Explanation

To solve the problem, we need to find the dimensions of the expression \( \frac{E J^2}{M^5 G^2} \), where \( E \) is energy, \( M \) is mass, \( J \) is angular momentum, and \( G \) is the gravitational constant. We will start by writing down the dimensions of each of these quantities.

### Step 1: Write down the dimensions of each variable

1. **Energy (E)**: The dimensional formula for energy is given by:
\[ [E] = [M][L^2][T^{-2}] = M L^2 T^{-2} \]

2. **Mass (M)**: The dimensional formula for mass is:
\[ [M] = M \]

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

4. **Gravitational Constant (G)**: The dimensional formula for 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 \( \frac{E J^2}{M^5 G^2} \).

\[ \frac{E J^2}{M^5 G^2} = \frac{(M L^2 T^{-2}) (M L^2 T^{-1})^2}{M^5 (M^{-1} L^3 T^{-2})^2} \]

### Step 3: Calculate \( J^2 \) and \( G^2 \)

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

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

### Step 4: Substitute back into the expression

Now substituting \( J^2 \) and \( G^2 \) back into the expression:
\[ \frac{E J^2}{M^5 G^2} = \frac{(M L^2 T^{-2}) (M^2 L^4 T^{-2})}{M^5 (M^{-2} L^6 T^{-4})} \]

### Step 5: Simplify the expression

Now we simplify the expression step by step:
1. The numerator becomes:
\[ M^{1+2} L^{2+4} T^{-2-2} = M^3 L^6 T^{-4} \]

2. The denominator becomes:
\[ M^5 (M^{-2} L^6 T^{-4}) = M^{5-2} L^6 T^{-4} = M^3 L^6 T^{-4} \]

Now we have:
\[ \frac{M^3 L^6 T^{-4}}{M^3 L^6 T^{-4}} = 1 \]

### Step 6: Conclusion

The result is dimensionless, which means it has no dimensions. Therefore, the final answer is that the expression \( \frac{E J^2}{M^5 G^2} \) is dimensionless.