Question

What must be the units for the gravitational constant G in order for gravitational force to have units of
newtons?

N x m^2/kg^2

N x m/kg^2

N x m/kg

N x m^3/kg^3

Answer 1

The gravitational force \( F \) between two masses is given by Newton's law of gravitation:

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

Where:
- \( F \) is the gravitational force in newtons (N),
- \( G \) is the gravitational constant,
- \( m_1 \) and \( m_2 \) are the masses in kilograms (kg),
- \( r \) is the distance between the centers of the two masses in meters (m).

To find the units of the gravitational constant \( G \), we can rearrange the formula:

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

Now we need to express the units for \( G \):

- The units of force \( F \) are newtons (N).
- The units of distance \( r \) are meters (m), thus \( r^2 \) will have units of \( m^2 \).
- The units for masses \( m_1 \) and \( m_2 \) are kilograms (kg), so their product \( m_1 m_2 \) will have units of \( kg^2 \).

Substituting these units into the expression for \( G \):

\[
[G] = \frac{[F][r^2]}{[m_1][m_2]} = \frac{N \cdot m^2}{kg \cdot kg} = \frac{N \cdot m^2}{kg^2}
\]

Thus, the units for the gravitational constant \( G \) must be:

\[
\text{N} \cdot \text{m}^2/\text{kg}^2
\]

The correct answer is therefore:

**N x m²/kg²**