What is the gravitational potential of the field of a body…

What is the gravitational potential of the field of a body of mass $m = 1\ \mathrm{kg}$ at a distance $r = 6.67\ \mathrm{m}$?

Explanation of Method

Potential of a 1 kg body.

Step 1: Idea

The gravitational potential at a point is the potential energy that 1 kg would have there. It is also equal to minus the work needed per kilogram to carry a body from that point to infinity, where the potential is taken to be zero. For a point mass (or a uniform sphere, outside it):

\[ \varphi = -\gamma\frac{m}{r}. \]

The minus sign expresses that gravity is attractive: we must do work to pull a body away, so the energy near the mass is lower than at infinity.

Step 2: Numbers

The distance was chosen so that the numbers cancel neatly: $\varphi = -\frac{6.67 \cdot 10^{-11} \cdot 1}{6.67}$.

\[\varphi = -1 \cdot 10^{-11}\ \mathrm{J/kg}\]

So a 1 kg test body at this point has a potential energy of $-10^{-11}\ \mathrm{J}$, and $10^{-11}\ \mathrm{J}$ of work would free it from the attraction completely.

Check

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

Sign: zero at infinity, so the potential near a mass is negative $\checkmark$

$\varphi$ $r\ (\mathrm{m})$ $2$ $4$ $6.67$ $8$ $-10^{-11}$ $\varphi(6.67\ \mathrm{m}) = -10^{-11}\ \mathrm{J/kg}$ $\varphi = -\frac{\gamma m}{r}$

The potential of a point mass: negative everywhere, rising toward zero far away.

Result

\[\boxed{\,\varphi = -1 \cdot 10^{-11}\ \mathrm{J/kg}\,}\]