Orbital radius and period do not depend on satellite mass
Since m cancels in GMm/r² = mrω², v, ω and T depend only on M and r. Two satellites with the same period must have the same radius, whatever their masses. Doubling a satellite's mass at fixed period leaves the radius unchanged.
What this actually means
The trap version is almost always MCQ: 'a 24-hour satellite is replaced by one of twice the mass, also with a 24-hour period, find the ratio of the new radius to the old.' The answer is 1 : 1, and the reason is that m cancelled.
It feels wrong because a heavier satellite is pulled harder. It is, but it also needs proportionally more centripetal force to turn on the same circle. The two effects scale identically with m, so they cancel exactly.
There is only ONE geostationary radius, for the same reason. Every geostationary satellite ever launched, from a small comms relay to a heavy weather platform, sits on the same 4.23 × 10⁷ m circle.
Be careful where mass DOES matter: KE = GMm/2r, GPE = −GMm/r and TE = −GMm/2r are all proportional to m, as is the gravitational force itself. Kinematic quantities are mass-free, energy quantities are not.
Answer template that scores: 'Since the satellite's mass cancels from GMm/r² = mrω², the period depends only on the orbital radius and the mass of the Earth. Same period means same radius, so the ratio is 1 : 1.'
Assuming a heavier satellite needs a different radius for the same period.
Prove it — watch it be true
- Set an orbit and note its period, then change the satellite mass and confirm the period readout does not move.
- Check the point sits at the same place on the T² vs r³ line regardless of satellite mass.
- Now open the energy bars and confirm KE, PE and TE all DO scale with satellite mass.