Describe gravitational potential energy in a radial field and relate work done to changes in field energy.

What this dot point is asking

This dot point extends gravity from the uniform field near the surface to the full radial field around a planet or star, where field strength and potential energy depend on distance.

Field lines and the radial field

A gravitational field is represented by field lines that point toward the mass creating the field. Around a point mass or sphere the lines are radial, spreading out evenly in all directions. Where the lines are closer together the field is stronger, so the field is strongest near the surface and weakens with distance following the inverse-square law $g = \dfrac{GM}{r^2}$.

Near the surface, over distances small compared with the planet's radius, the lines are almost parallel and evenly spaced, which is why we treat the surface field as uniform and use $g = 9.8\ \text{N kg}^{-1}$.

Gravitational potential energy in a radial field

Close to the surface we use $E_p = mgh$, but this assumes constant $g$. Over large distances $g$ changes, so the correct expression for the gravitational potential energy of a mass $m$ at distance $r$ from the centre of mass $M$ is:

The energy is negative because the zero of potential energy is defined at infinite separation, where the masses no longer interact. As you move a mass closer, the energy becomes more negative; as you move it away, it rises toward zero. Bringing the masses together releases energy, which is why falling objects gain kinetic energy.

Work done moving through the field

The work needed to move a mass between two points equals the change in its gravitational potential energy:

Moving away from the planet ($r_2 > r_1$) makes $E_p$ less negative, so $\Delta E_p$ is positive and you must do work. Moving inward releases energy. This is the energy a rocket must supply to climb to a higher orbit.

Field-distance graphs

If you plot gravitational field strength $g$ against distance $r$, the area under the graph between two radii gives the change in gravitational potential per unit mass. Because the field follows an inverse-square curve, this area is found by integration or by reading values, and it confirms that most of the energy to escape a planet is spent close to the surface where the field is strongest.

In the exam, check the scale of the height change. If it is small compared with the planet's radius, $mgh$ is fine. If the object moves a large distance or out to space, switch to $E_p = -\dfrac{GMm}{r}$ and work with the change in energy.