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A geostationary satellite orbits Earth at a distance r from its centre. If Earth's rotational angular velocity were to double, at what distance from the centre would a satellite now have to orbit to remain geostationary?
- r/2
- r/(2*sqrt(2))
- r/(4)^(1/3)
- r/(2)^(1/3)
Correct answer: r/(4)^(1/3)
Solution
For a geostationary orbit, orbital angular velocity equals Earth's spin omega. From gravity = centripetal: G*Mₑ/r² = omega²*r, so r³ = G*Mₑ/omega², i.e. r proportional to omega^(-2/3). If omega -> 2*omega, then r' = r*(omega/(2*omega))^(2/3) = r*(1/2)^(2/3) = r/2^(2/3) = r/(4)^(1/3).
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