Explain apparent weightlessness and apply Kepler's laws to the orbits of satellites and planets.

What this dot point is asking

This dot point applies gravity and circular motion to real orbits, and explains the often misunderstood idea of weightlessness in space.

Apparent weightlessness

Weight is the gravitational force $W = mg$, and it does not disappear in orbit. At the altitude of the International Space Station, gravity is still about 90 per cent of its surface value. What changes is the apparent weight, which is the normal force a surface pushes back with.

In orbit the station and everything in it are in continuous free fall toward Earth, accelerating at the local $g$. Because the floor falls with the astronaut, it pushes with zero normal force, and with no normal force there is nothing to sense as weight. This is true weightlessness only in deep space far from any mass; in orbit it is apparent weightlessness caused by free fall.

Low Earth and geostationary orbits

The orbital speed $v = \sqrt{\dfrac{GM}{r}}$ shows that lower orbits are faster. A low Earth orbit has a period of about 90 minutes. A geostationary orbit is special: its period is exactly one sidereal day, so the satellite stays above a fixed point on the equator, ideal for communications. Setting $T = 86400\ \text{s}$ in Kepler's third law gives a radius of about $4.2 \times 10^7\ \text{m}$, roughly 36000 km above the surface.

Kepler's three laws

Kepler described planetary orbits before Newton explained them:

1. The law of ellipses: each planet moves in an ellipse with the Sun at one focus. For most school problems we approximate orbits as circles.
2. The law of equal areas: a line from the Sun to a planet sweeps out equal areas in equal times, so a planet moves faster when closer to the Sun.
3. The law of periods: the square of the orbital period is proportional to the cube of the orbital radius.

The third law follows directly from equating gravity to the centripetal force:

so $\dfrac{T^2}{r^3}$ is the same constant for every satellite of a given central body, and you can compare two orbits without knowing $G$ or $M$.

In the exam, when comparing orbits use Kepler's third law as a ratio so the unknown constants drop out. When asked about weightlessness, state clearly that gravity still acts and that the sensation comes from the absence of a normal force during free fall.