For a satellite, its period [math: T] is related to its radius of orbit [math: R] by the equation [math: R3=KT2] , where [math: K] is a constant. For an earth satellite, the equation becomes [math: R3=4KT2] If [math: G] is the gravitational constant and [math: M] is the mass of the earth, which one of the following is equal to [math: K] for any earth satellite?

Question

BlackTom AI · Accepted Answer

Kepler's third law (derived from Newton's law of gravitation and circular motion) gives the key relation between orbital period and radius.

1) Relevant knowledge points and concepts
- Gravitational force provides the centripetal force for a satellite: G M m / R^2 = m v^2 / R.
- Orbital speed v for a circular orbit: v = sqrt(G M / R).
- Period T is the circumference divided by speed: T = 2πR / v.
- Combining these yields the standard form T^2 = (4π^2 / (G M)) R^3, so R^3 is proportional to T^2 and the proportionality constant K can be identified.

2) Step-by-step reasoning
Start with force balance:
G M m / R^2 = m v^2 / R.
Cancel m and rearrange:
v^2 = G M / R.
Express T in terms of v:
T = 2πR / v.
Square T:
T^2 = (4π^2 R^2) / v^2.

3) Intermediate calculation steps
Substitute v^2 = G M / R into T^2:
T^2 = (4π^2 R^2) / (G M / R) = (4π^2 R^3) / (G M).
Rearrange to express R^3 in terms of T^2:
R^3 = (G M / 4π^2) T^2.

4) Why the correct answer is correct
Comparing R^3 = K T^2 with R^3 = (G M / 4π^2) T^2 shows K = G M / (4π^2). This matches option a.

5) Why the other options are incorrect
- b (4π^2 / GM): This is the reciprocal of the correct K and would produce T^2 = (GM/4π^2) R^3, which is dimensionally and algebraically wrong.
- c (GM / 16π^2): This differs by a factor of 4 (should be 4π^2 in denominator, not 16π^2), so it gives the wrong proportionality constant.
- d (2 / √(GM)): This expression has incorrect dimensions (K must have dimensions of length^3 / time^2) and is not obtained from the algebra above.

6) Conclusion
Therefore K = GM / (4π^2), which corresponds to option (a). This confirms the provided correct answer.

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