The strength of the gravitational field at a certain point…

The strength of the gravitational field at a certain point is $G = 9.8\ \mathrm{N/kg}$, and the gravitational potential there is $\varphi = -67\ \mathrm{MJ/kg}$. A body of mass $m = 2\ \mathrm{kg}$ is placed at this point. (a) What gravitational force acts on the body? (b) What gravitational potential energy does the body have?

Step 1: Idea

Field strength and potential are properties of a point in the field, not of the body placed there. They are defined \"per kilogram\": the field strength is the force on each kilogram, and the potential is the potential energy of each kilogram. To get the force and the energy of a particular body we simply multiply by its mass.

\[ \vec G = \frac{\vec F}{m}, \]

\[ \varphi = \frac{E_p}{m}. \]

Step 2: (a) Force

From the definition of field strength: $F = mG = 2\ \mathrm{kg} \cdot 9.8\ \mathrm{N/kg}$.

\[F = 19.6\ \mathrm{N}\]

The force points in the direction of the field, i.e. toward the body that creates the field.

Step 3: (b) Potential energy

The potential must be in SI units: $-67\ \mathrm{MJ/kg} = -67 \cdot 10^6\ \mathrm{J/kg}$. Then

\[ E_p = m\varphi = 2 \cdot (-67 \cdot 10^6). \]

\[E_p = -1.34 \cdot 10^8\ \mathrm{J}\]

The energy is negative because the zero of potential energy is taken infinitely far away. The body is \"bound\" in the field: to carry it away to infinity we would have to supply $1.34 \cdot 10^8\ \mathrm{J}$.

Check

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

Plausibility: for a point mass $ |\varphi|/G = r $, here $ 6.8 \cdot 10^6\ \mathrm{m} $, close to the Earth's radius, so these are roughly the values at the Earth's surface $\checkmark$