If E, M, L and G denote energy, mass, angular momentum and gravitational constant repectively then the quantity `(E^(2)L^(2)//M^(5)G^(2))` has the dimensions of :-

Options

A. angle
B. length
C. mass
D. None of these

Explanation

To solve the problem, we need to determine the dimensions of the quantity \(\frac{E^2 L^2}{M^5 G^2}\) using the known dimensions of energy (E), mass (M), angular momentum (L), and the gravitational constant (G).

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

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

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

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 substitute the dimensions into the expression \(\frac{E^2 L^2}{M^5 G^2}\):

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

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

3. **Calculate \(M^5\)**:
\[ [M^5] = M^5 \]

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

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

### Step 4: Simplify the expression
1. **Numerator**:
\[ [\text{Numerator}] = M^2 L^4 T^{-4} \cdot M L^2 T^{-2} = M^{2+1} L^{4+2} T^{-4-2} = M^3 L^6 T^{-6} \]

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

### Step 5: Final dimension calculation
Now we can simplify:
\[ \frac{M^3 L^6 T^{-6}}{M^3 L^6 T^{-4}} = M^{3-3} L^{6-6} T^{-6+4} = M^0 L^0 T^{-2} = T^{-2} \]

### Conclusion
The dimensions of the quantity \(\frac{E^2 L^2}{M^5 G^2}\) are \(T^{-2}\), which corresponds to the dimension of acceleration or frequency.

### Final Answer
The quantity \(\frac{E^2 L^2}{M^5 G^2}\) has the dimensions of **none of the given options** (angle, length, mass).

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