Newton’s Law of Gravitation

Newton’s law states that the gravitational force between two point masses is proportional to the product of their masses and inversely proportional to the square of the distance between their centres.

The constant is $G=6.67\times 10^{-11}\mathrm{N}{\mathrm{m}}^2\mathrm{k}{\mathrm{g}}^{-2}$. Its small size is why you do not notice the pull between everyday objects.

Spherical objects such as planets behave as if all their mass were concentrated at their centre, so $r$ is measured centre to centre.

A $\text{gravitational field}$ is a region where a mass experiences a force. The field strength $g$ at a point is the force per unit mass on a small test mass placed there, in $\mathrm{N}\mathrm{k}{\mathrm{g}}^{-1}$.

Field lines point in the direction a mass would be pulled — always $\text{towards}$ the attracting mass. Near Earth’s surface the field is effectively $\text{uniform}$, with parallel, equally spaced lines, and $g=9.81\mathrm{N}\mathrm{k}{\mathrm{g}}^{-1}$.

On a large scale, the field around a planet is $\text{radial}$, with lines converging on its centre.

Dividing Newton’s law by the test mass gives the field strength at distance $r$ from the centre of a mass $M$.

Outside the planet, $g$ falls as $1/r^2$. Doubling the distance from the centre reduces it to a quarter.

Treating $g$ as constant is only valid for height changes that are small compared with the planet’s radius. A graph of $g$ against $r$ is a steep inverse square curve that flattens towards zero far from the planet.

Both are inverse square fields: $F=G{m}_1{m}_2/r^2$ and $F=k{Q}_1{Q}_2/r^2$. Both have radial fields for point sources and uniform fields in special cases.

Gravity acts on $\text{mass}$ and is always attractive. The electric force acts on $\text{charge}$ and can attract or repel.

Gravity is vastly weaker. It dominates astronomy only because large bodies carry huge mass but almost no net charge — positive and negative charges cancel, but masses always add.