Revise Gravitational potential for Cambridge 9702 (13.4)

Gravitational potential, symbolised by $\phi (phi)$, tells us the work an external force must perform to bring a unit mass (a 1 kg mass) from an infinitely far point (where gravity's influence is zero) to a specific location within a gravitational field. It’s a scalar quantity, meaning it only has magnitude, not direction, and is measured in Joules per kilogram (J kg^-1).

$\phi = -\frac{GM}{r}$

Gravitational forces are always attractive. As a mass moves from infinity towards a gravitating body, the field itself does work, releasing energy. Since gravitational potential is defined as the work done by an external agent to bring a unit mass from infinity, and energy is released, this work is negative. Hence, potential is always negative at any finite distance from a source mass.

An equipotential surface is a surface on which the gravitational potential ($\phi$) is constant at every point. For an isolated point mass or a spherical body, these surfaces are concentric spheres. A key property is that no work is done in moving a mass along an equipotential surface, because the change in potential energy ($mΔ\phi$) is zero.

Gravitational field strength (g) and gravitational potential ($\phi$) are intimately related. The field strength is the negative of the potential gradient. This means the field strength at a point is equal to how steeply the potential changes with distance at that point.

$g = -\frac{\Delta\phi}{\Delta r}$

This relationship is analogous to a topographical map: the gravitational field strength 'g' is like the steepness of a hill, while the gravitational potential 'φ' is like the altitude. A steep hill (large 'g') corresponds to rapidly changing altitude (large potential gradient).

While gravitational potential ($\phi$) is for a unit mass, gravitational potential energy (E_p) is the total energy an object of mass 'm' possesses due to its position in a gravitational field. It's the total work an external agent must do to bring that specific mass 'm' from infinity to its current location.

$E_p = -\frac{GMm}{r}$