Question

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

N.m²/kg

Nm²/kg²

N.m/kg

N.m/kg²

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 determine the units of \( G \), we need to rearrange the formula to express \( G \):

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

Now we can substitute in the units:

- The unit of force \( F \) is newtons (N).
- The unit of distance \( r \) is meters (m), so \( r^2 \) has units of m².
- The unit of mass \( m_1 \) and \( m_2 \) is kilograms (kg).

Substituting these units into the equation gives us:

\[
G = \frac{N \cdot m^2}{kg \cdot kg} = \frac{N \cdot m^2}{kg^2}
\]

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

\[
\text{Nm}^2/\text{kg}^2
\]

Now, looking at the options provided, the correct choice that matches the units of \( G \) is:

**Nm²/kg²**