Determine the strength of the gravitational field of a body…

Determine the strength of the gravitational field of a body of mass $m = 1\ \mathrm{kg}$ at a distance $r = 10\ \mathrm{m}$.

Step 1: Idea

The field strength at a point is the force that the field would exert on a test body of 1 kg placed there. Put a test mass $m_0$ at distance $r$ from the body; Newton's law gives the force, and dividing by $m_0$ gives the field strength:

\[ F = \gamma\frac{m\,m_0}{r^2}, \]

\[ G = \frac{F}{m_0} = \gamma\frac{m}{r^2}. \]

The test mass cancels, as it must: the field belongs to the body $m$ alone.

Step 2: Numbers

\[ G = \frac{6.67 \cdot 10^{-11} \cdot 1}{10^2} = \frac{6.67 \cdot 10^{-11}}{100}. \]

\[G = 6.67 \cdot 10^{-13}\ \mathrm{N/kg}\]

The field points toward the 1 kg body. It is about $1.5 \cdot 10^{13}$ times weaker than the Earth's field at the surface ($9.81\ \mathrm{N/kg}$), which is why we never notice the attraction of everyday objects.

Check

Units: $ \mathrm{N\,m^2/kg^2} \cdot \mathrm{kg/m^2} $ $= \mathrm{N/kg}\ \checkmark $

Doubling the distance would reduce the field four times (inverse-square law) $\checkmark$