Newton's Law of Gravitation
Every object with mass pulls on every other object with mass, all the time, everywhere in the universe — you're tugging gently on your phone right now, and it's tugging back. Newton's law of gravitation packages that into one compact equation, underneath planetary orbits, satellite motion, tides, and the whole of gravitational field theory you're about to build in this topic.
The law itself
Newton's law of gravitation states that any two point masses attract each other with a force proportional to the product of their masses and inversely proportional to the square of the distance between them:
where $F$ is the magnitude of the mutual attractive force between the two masses (in newtons), $m_1$ and $m_2$ are the two point masses (in kilograms), $r$ is the distance between their centres (in metres), and $G$ is the gravitational constant:
$G$ is one of the fundamental constants of nature — the same value everywhere in the universe, as far as every measurement to date can tell. It's also notoriously one of the hardest constants to measure precisely, since gravity is such an extraordinarily weak force between laboratory-sized masses; the accepted value carries more uncertainty than almost any other fundamental constant in physics.
By Newton's third law, the force $m_1$ exerts on $m_2$ is equal in magnitude and opposite in direction to the force $m_2$ exerts on $m_1$ — both masses experience the same size of pull, directed toward each other, regardless of how different they are. A person standing on the Earth pulls on the Earth just as hard as the Earth pulls on them; it's only because the Earth's mass is so enormous that its resulting acceleration is utterly negligible compared to the person's.
Where does this law come from?
Unlike many results in A-level physics, Newton's law of gravitation isn't derived from some deeper principle — it's the fundamental starting point, an empirical law inferred by Newton from observations (particularly the motion of the Moon and planets, matched against Kepler's earlier empirical laws). You're not expected to derive it; you're expected to know it, use it correctly, and understand the reasoning behind its structure.
That said, the structure isn't arbitrary. The $\frac{1}{r^2}$ dependence has a natural geometric flavour, closely connected to how anything spreading out from a point source gets diluted over the surface of an expanding sphere — the same inverse-square pattern shows up for light intensity from a point source and the electric field around a point charge. Inverse-square laws recur throughout physics wherever something spreads uniformly outward from a point in three-dimensional space.
What the inverse-square law means in practice
Because $F \propto \dfrac{1}{r^2}$, gravitational force is extremely sensitive to distance. Doubling the separation doesn't halve the force — it cuts it to a quarter, since $\left(\dfrac{1}{2}\right)^2 = \dfrac{1}{4}$. Tripling the distance cuts the force to a ninth; ten times the distance weakens it by a factor of 100.
The reverse is just as dramatic: halve the separation and the force quadruples. This is why gravitational effects near a very compact, close object (imagine standing at the surface of a neutron star) can be catastrophically larger than the same mass would produce at ordinary planetary distances — the force scales incredibly steeply with proximity.
A common slip is to treat $F \propto \dfrac{1}{r^2}$ as if it were $F \propto \dfrac{1}{r}$ — halving $r$ and simply doubling $F$. Always square the distance ratio first. If $r$ becomes $\dfrac{r}{n}$, the force becomes $n^2 F$, not $nF$.
Gravity is always attractive
Every pair of masses in the universe pulls toward each other — no configuration of ordinary matter produces a repulsive gravitational force. This follows from mass only coming in one "sign": there's no negative mass to flip the direction of the force, the way a negative electric charge flips the direction of an electric force. Every mass gravitationally attracts every other mass. Gravity can be balanced by other forces (how you stand still on the ground, held up by the floor's contact force), but it can never itself push two masses apart.
Newton's law of gravitation, $F = \dfrac{Gm_1m_2}{r^2}$, and Coulomb's law for the electric force between two point charges, $F = \dfrac{Qq}{4\pi\varepsilon_0 r^2}$, share exactly the same inverse-square structure — a constant, multiplied by the product of two "source" quantities, divided by $r^2$. But there's one crucial difference: mass $m_1m_2$ is always positive, so the gravitational force is always attractive. Charge $Qq$ can be positive or negative depending on the two charges' signs, so the Coulomb force can be either attractive or repulsive. Don't let the matching algebra fool you into thinking the physics behaves the same way in both cases.
The two forces also differ enormously in strength. For two protons, electric repulsion is roughly $10^{36}$ times stronger than their mutual gravitational attraction. Gravity is negligible at the scale of individual particles, which is why atomic and nuclear physics can almost always ignore it. Gravity dominates at astronomical scales for two reasons together: mass always adds up (unlike charge, where equal positive and negative charge in ordinary matter largely cancels), and astronomical objects contain vast quantities of it. A planet or star carries no significant net electric charge, so its far larger mass determines the force it exerts.
Treating spheres as point masses
Newton's law as written applies exactly to point masses — objects with no physical extent, or small enough compared to $r$ that their size can be ignored. Real astronomical objects like the Earth, the Moon and the Sun are obviously not points; they're enormous spheres. Fortunately, a result going back to Newton himself (its full proof needs calculus well beyond A-level, using what's now called the shell theorem) shows that a uniform spherical mass behaves gravitationally, from any point outside its surface, exactly as though the entire mass were concentrated at its centre.
You don't need to prove this at A-level — just use it. It's why you can calculate the gravitational force between the Earth and the Moon by treating each as a point mass at its centre, plugging the centre-to-centre distance into $r$, even though both are extended spheres. The same reasoning lets you calculate your own weight using the distance from you to the centre of the Earth (essentially the Earth's radius, since you're negligibly above the surface by comparison), not to the nearest point on the surface.
The shell theorem's "acts as a point mass at the centre" result applies to points outside a uniform sphere's surface (or, for a hollow shell, also on and outside its surface). It does not apply to points inside a uniform solid sphere, where the effective enclosed mass — and hence the field — behaves differently. A-level questions on gravitational fields inside a planet are rare, but if you ever see one, don't reach for $\frac{GMm}{r^2}$ using the full mass $M$.
Worked example: the Earth–Moon system
Find the gravitational force of attraction between the Earth and the Moon, given:
- Mass of Earth, $m_1 = 5.97\times10^{24}\,\text{kg}$
- Mass of Moon, $m_2 = 7.35\times10^{22}\,\text{kg}$
- Earth–Moon distance (centre to centre), $r = 3.84\times10^{8}\,\text{m}$
Notice that this same force of about $1.98\times10^{20}\,\text{N}$ acts on both bodies — it's what keeps the Moon in orbit around the Earth (or, more precisely, what keeps both bodies orbiting their common centre of mass), and it's also the underlying cause of tides on Earth, since the Moon pulls slightly more strongly on the near side of the Earth than on the far side.
Key takeaways
- Newton's law of gravitation: $F = \dfrac{Gm_1m_2}{r^2}$, giving the mutual attractive force between two point masses $m_1$ and $m_2$ separated by distance $r$.
- $G = 6.67\times10^{-11}\,\text{N m}^2\text{kg}^{-2}$ is the gravitational constant — the same everywhere in the universe.
- The law is inverse-square: doubling $r$ cuts $F$ to a quarter; halving $r$ quadruples $F$.
- Gravity is always attractive — there's no negative mass, so unlike Coulomb's law, gravitational force can never be repulsive.
- Gravity is enormously weaker than the electric force between individual particles (around $10^{36}$ times weaker for two protons), but dominates at astronomical scales because mass never cancels, while charge in bulk matter usually does.
- A uniform sphere (such as a planet or star) can be treated as a point mass at its centre for any point outside its surface — this is why $F = \dfrac{GMm}{r^2}$ can be applied directly to bodies like the Earth and Moon.