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Rearranging Kepler's third law as r^3/T^2 = GM/(4 pi^2) shows that this ratio should be the same for every satellite orbiting a given parent body. What can this constant ratio be used to find?
AThe radius of the satellite itself
BThe mass of the parent body being orbited
CThe exact shape of the satellite's orbit
DThe gravitational constant G for that system only
Answer & Solution
Correct answer: B. The mass of the parent body being orbited
1. Solving T^2 = (4 pi^2/GM) r^3 for the ratio r^3/T^2 gives r^3/T^2 = GM/(4 pi^2).
2. Because G and 4 pi^2 are fixed constants, this ratio depends only on the parent body's mass M, and it should be the same for every satellite of that parent.
3. If r and T are known for even one satellite, this relationship can be solved to determine the mass M of the body it orbits.
4. The ratio does not directly give the satellite's own radius, its orbital shape, or a value of G specific to that system, since G is a universal constant, not one determined separately for each orbit.
_Source: OpenStax College Physics (CC BY 4.0), Ch 6 "Uniform Circular Motion and Gravitation", section 6.6 Satellites and Kepler's Laws: An Argument for Simplicity_
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