What is the dimensional formula of gravitational constant ?

Options

A. `[ML^(2)T^(-2)]`
B. `[ML^(-1)T^(-1)]`
C. `[M^(-1)L^(3)T^(-2)]`
D. None of these

Answer

To find the dimensional formula of the gravitational constant \( g \), we start from Newton's law of gravitation, which states:

\[ F = \frac{G m_1 m_2}{r^2} \]

Where:
- \( F \) is the gravitational force,
- \( G \) is the gravitational constant,
- \( m_1 \) and \( m_2 \) are the masses of the two objects,
- \( r \) is the distance between the centers of the two masses.

1. **Rearranging the formula for \( G \)**:
We can rearrange the equation to solve for \( G \):

\[ G = \frac{F r^2}{m_1 m_2} \]

2. **Substituting the dimensions**:
We know the dimensions of the quantities involved:
- The dimension of force \( F \) is given by \( [F] = [M][A] = [M][L][T^{-2}] \).
- The dimension of distance \( r \) is \( [L] \).
- The dimension of mass \( m_1 \) and \( m_2 \) is \( [M] \).

Substituting these into the equation for \( G \):

\[ G = \frac{[M][L][T^{-2}] \cdot [L^2]}{[M] \cdot [M]} = \frac{[M][L^3][T^{-2}]}{[M^2]} \]

3. **Simplifying the expression**:
Now, we can simplify the expression:

\[ G = [M^{1-2}][L^3][T^{-2}] = [M^{-1}][L^3][T^{-2}] \]

4. **Final dimensional formula**:
Thus, the dimensional formula of the gravitational constant \( G \) is:

\[ [G] = [M^{-1} L^3 T^{-2}] \]

### Summary
The dimensional formula of the gravitational constant \( G \) is \( [M^{-1} L^3 T^{-2}] \).