Quick questions on Orbital motion, Kepler's third law and satellite energy (QCE Physics Unit 3)

For a satellite of mass $m$ in a circular orbit of radius $r$ around a central body of mass $M$, the gravitational force supplies all of the centripetal force:

The kinetic energy of a satellite in a circular orbit of radius $r$ is:

Expect a satellite or moon table (radii and periods, sometimes a missing column) with a question asking you to verify Kepler's third law or extract $M$ of the central body. Alternatively, a stimulus showing the orbital energies as a function of radius with questions on the virial theorem.

A practical IA2 on orbits is hard to engineer directly, but a frequent design is the simple pendulum used to measure local $g$, then comparing the inferred $G M_E / R_E^2$ against the textbook value. The orbital framework provides the EA-level theory for the Unit 3 justification.

State Kepler's third law for satellites of a central mass $M$. [2 marks]

Calculate the orbital speed of a $400 \text{ km}$ altitude satellite ($r = 6.77 \times 10^6 \text{ m}$, $M_E = 5.97 \times 10^{24} \text{ kg}$). [3 marks]

ANSTO tracks a geostationary satellite at $r = 4.22 \times 10^7 \text{ m}$. (a) Calculate the orbital speed and period. (b) Verify against Kepler's third law.