Gravitational Fields

Two masses attract each other across empty space with nothing visibly connecting them. The field is the way physics describes that: a region in which a mass experiences a force, with a strength and direction at every point.

Gravitational field strength g is defined as the force per unit mass. That definition is what makes it useful — it strips out the mass of whatever happens to be sitting there and describes the field itself. It also explains the coincidence that g = 9.81 N kg⁻¹ and the acceleration of free fall is 9.81 m s⁻²: they are the same quantity, since F = mg and F = ma give a = g.

G

the gravitational constantthe same everywhere in the universe

r

separation of the centresnot the surface-to-surface distance

M

the mass creating the fieldthe small mass in it does not appear in g

Gravitational potential φ is the work done per unit mass in bringing a mass from infinity to that point. The zero is placed at infinity, because that is the only place where masses genuinely stop interacting.

Since gravity is always attractive, it does the work of pulling a mass inwards — nothing has to be supplied. So the work done on the mass is negative, and the potential is negative everywhere, approaching zero only at infinity. A negative potential is not an error; it is the whole structure of a gravitational well.

Field and potential follow different powers of r, which is the distinction that most needs holding onto: field goes as 1/r² and potential as 1/r. They are linked by calculus — the field is the negative gradient of the potential.

φ

gravitational potentialwork per unit mass from infinity, in J kg⁻¹

E_p

potential energyφ multiplied by the mass placed there, in J

zero at infinity

the chosen referencewhich forces every finite value to be negative

Switch to Both and drag r. Doubling the distance quarters the field but only halves the potential. The potential curve is the shallower of the two and lies entirely below the axis — that is the gravitational well.

A satellite in a circular orbit is accelerating, because its direction is constantly changing. The only force acting is gravity, so gravity must be exactly the centripetal force required. Setting those two expressions equal is the key step in almost every orbit question.

The mass of the satellite cancels immediately, which is why the orbital speed and period depend only on the radius and the mass of the central body. A heavy satellite and a light one at the same altitude orbit at the same speed, and astronauts float not because gravity is absent but because they and the station are falling together.

v

orbital speedindependent of the satellite mass

T

the orbital periodT² proportional to r³

geostationary

T = 24 hourswhich fixes r at about 42 000 km from the centre

The inverse square law says the field weakens with distance, yet every mechanics question treats g as a fixed 9.81. Both are correct, and the reason is a matter of scale.

The Earth's radius is about 6370 km. Climbing a 100 m building changes r by roughly one part in 64 000, and squaring that still leaves a change of about 0.003% — far below the precision of any school measurement. Over the small heights of ordinary mechanics the field is genuinely uniform for practical purposes.

Over larger distances the variation becomes impossible to ignore. At the altitude of the International Space Station g has fallen to about 8.7 N kg⁻¹, roughly 89% of its surface value, and at geostationary height it is under 0.25 N kg⁻¹. This is why a satellite question must always use the inverse square law rather than the constant value.