The gravitational force F between two masses `m_(1) and m_(2)` separeted by a distance r is given by
`F = (Gm_(1)m_(2))/(r^(2))`
Where G is the universal gravitational constant. What are the dimensions of G ?

Options

A. `M^(-1)L^(2)T^(-3)`
B. `M^(-1)L^(3)T^(-2)`
C. `ML^(3)T^(-2)`
D. `ML^(2)T^(-3)`

Answer

To find the dimensions of the universal gravitational constant \( G \) from the gravitational force equation, we can follow these steps:

### Step-by-Step Solution:

1. **Start with the gravitational force equation**:
\[ F = \frac{G m_1 m_2}{r^2} \]

2. **Rearrange the equation to solve for \( G \)**:
\[ G = \frac{F r^2}{m_1 m_2} \]

3. **Identify the dimensions of each variable**:
- The dimensional formula for force \( F \) is:
\[ [F] = M L T^{-2} \]
- The dimensional formula for distance \( r \) is:
\[ [r] = L \]
- The dimensional formula for masses \( m_1 \) and \( m_2 \) is:
\[ [m_1] = [m_2] = M \]

4. **Substitute the dimensions into the equation for \( G \)**:
- The dimension of \( r^2 \) is:
\[ [r^2] = [L^2] \]
- Therefore, substituting into the equation for \( G \):
\[ [G] = \frac{[F] \cdot [r^2]}{[m_1] \cdot [m_2]} = \frac{(M L T^{-2}) \cdot (L^2)}{M \cdot M} \]

5. **Simplify the expression**:
\[ [G] = \frac{M L^3 T^{-2}}{M^2} = M^{-1} L^3 T^{-2} \]

6. **Final result**:
The dimensions of the universal gravitational constant \( G \) are:
\[ [G] = M^{-1} L^3 T^{-2} \]

### Summary:
The dimensions of the universal gravitational constant \( G \) are \( M^{-1} L^3 T^{-2} \).