Apply Newton's law of universal gravitation and the field model to orbital motion.

What this dot point is asking

Newton's law of universal gravitation

Newton proposed that any two point masses attract each other along the line joining them with a force proportional to the product of their masses and inversely proportional to the square of their separation:

Here $G = 6.67 \times 10^{-11}\ \text{N m}^2\,\text{kg}^{-2}$ is the universal gravitational constant and $r$ is the distance between the centres of the masses. The inverse-square dependence is crucial: doubling the separation reduces the force to a quarter.

The gravitational field

It is often more useful to describe gravity as a field. The gravitational field strength $g$ at a point is the force per unit mass placed there:

where $M$ is the mass creating the field. Near Earth's surface this evaluates to about $9.8\ \text{N kg}^{-1}$, which is the same number as free-fall acceleration in $\text{m s}^{-2}$. Field strength is a vector pointing toward the mass that creates it, and it falls off as the inverse square of distance from the centre.

Orbital motion

For a satellite in a stable circular orbit, gravity is the only force acting and it points toward the central body, so gravity provides exactly the centripetal force needed:

The satellite mass $m$ cancels, leaving the orbital speed:

A higher orbit means a slower speed. Substituting $v = \frac{2\pi r}{T}$ and rearranging gives Kepler's third law:

so $T^2$ is proportional to $r^3$ for all satellites of the same central body.

A geostationary satellite is a special case: it has a period of exactly one day so it stays above the same point on the equator. Setting $T = 86400\ \text{s}$ in Kepler's third law gives an orbital radius of about $4.2 \times 10^7\ \text{m}$.

When tackling orbital questions, decide first whether you need the field equation or the orbit equation. Use $g = \frac{GM}{r^2}$ for field strength and weight, and use $\frac{GMm}{r^2} = \frac{mv^2}{r}$ whenever the body is actually orbiting, since that is what links gravity to circular motion.