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. length
B. mass
C. time
D. angle

Answer

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 the gravitational constant (G).

### Step 1: Determine the dimensions of each variable.

1. **Energy (E)**:
The dimensions of energy are given by:
\[ [E] = [M][L^2][T^{-2}] = M^1 L^2 T^{-2} \]

2. **Mass (M)**:
The dimensions of mass are:
\[ [M] = M^1 \]

3. **Angular Momentum (J)**:
Angular momentum is defined as the product of mass, velocity, and radius. Its dimensions can be expressed as:
\[ [J] = [M][L][T^{-1}] \cdot [L] = M^1 L^2 T^{-1} \]

4. **Gravitational Constant (G)**:
The dimensions of the gravitational constant can be derived from the formula for gravitational force:
\[ F = \frac{G m_1 m_2}{r^2} \]
Rearranging gives:
\[ [G] = \frac{[F][L^2]}{[M]^2} = \frac{[M][L][T^{-2}][L^2]}{[M]^2} = M^{-1} L^3 T^{-2} \]

### Step 2: Substitute the dimensions into the expression.

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

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

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

### Step 3: Combine the dimensions in the expression.

Now we can substitute these into 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.

Now we simplify the dimensions:
\[ = \frac{M^{1+2} L^{2+4} T^{-2-2}}{M^5 M^{-2} L^6 T^{-4}} = \frac{M^3 L^6 T^{-4}}{M^{5-2} L^6 T^{-4}} = \frac{M^3 L^6 T^{-4}}{M^3 L^6 T^{-4}} = 1 \]

### Step 5: Conclusion.

The final result gives us:
\[ [M^0 L^0 T^0] = 1 \]
This indicates that the expression has no dimensions, which corresponds to an angle.

### Final Answer:
The dimensions of \( \frac{E J^2}{M^5 G^2} \) are dimensionless, indicating it can be associated with an angle.

---