SI units of G is `Nm^(2)kg^(-2)` .Which of the following can also be used as the SI unit of G?

Options

A. `m^(3)kg^(-1)s^(-2)`
B. `m^(2)kg^(-2)s^(-1)`
C. `mkg^(-3)s^(-1)`
D. `m^(2)kg^(-3)s^(-2)`

Explanation

To solve the problem of finding an alternative SI unit for the gravitational constant \( G \), we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Given Unit of \( G \)**:
The SI unit of \( G \) is given as \( \text{Nm}^2\text{kg}^{-2} \).

2. **Recall the Formula for Gravitational Force**:
The gravitational force \( F \) between two masses \( m_1 \) and \( m_2 \) separated by a distance \( r \) is given by:
\[ F = \frac{G m_1 m_2}{r^2} \]

3. **Rearranging the Formula to Isolate \( G \)**:
Rearranging the formula to solve for \( G \):
\[ G = \frac{F r^2}{m_1 m_2} \]

4. **Substituting SI Units**:
- The SI unit of force \( F \) is Newton (N).
- The SI unit of distance \( r \) is meter (m).
- The SI unit of mass \( m_1 \) and \( m_2 \) is kilogram (kg).

5. **Expressing the Units**:
Substitute the units into the equation for \( G \):
\[ G = \frac{\text{N} \cdot \text{m}^2}{\text{kg} \cdot \text{kg}} = \frac{\text{N} \cdot \text{m}^2}{\text{kg}^2} \]

6. **Convert Newton to Base SI Units**:
Recall that \( 1 \text{ N} = 1 \text{ kg} \cdot \text{m/s}^2 \). Therefore, we can substitute this into our expression for \( G \):
\[ G = \frac{(1 \text{ kg} \cdot \text{m/s}^2) \cdot \text{m}^2}{\text{kg}^2} = \frac{1 \text{ kg} \cdot \text{m}^3}{\text{kg}^2 \cdot \text{s}^2} \]

7. **Simplifying the Expression**:
Simplifying the expression gives:
\[ G = \frac{\text{m}^3}{\text{kg} \cdot \text{s}^2} \]

8. **Final Result**:
Thus, an alternative SI unit for \( G \) can be expressed as:
\[ G = \text{m}^3 \text{kg}^{-1} \text{s}^{-2} \]

### Conclusion:
The alternative SI unit for the gravitational constant \( G \) is \( \text{m}^3 \text{kg}^{-1} \text{s}^{-2} \).