The universal gravitation law and Newton's second law allow us to find that the answer for the relation of the rotation periods of the satellites is:
= 2.83
The universal gravitation law states that the force between two bodies is proportional to their masses and inversely proportional to their distance squared
Where G is the universal gravitational constant (G = 6.67 10⁻¹¹ ), F the force, m and m the masses of the bodies and r the distance between them
Newton's second law states that force is proportional to the mass and acceleration of bodies
F = m a
Where F is the force, m the mass and the acceleration
In this case the body is the satellites of Jupiter and the planet,
Suppose the motion of the satellites is circular, then the acceleration is centripetal
a = r
Where v is the speed of the satellite and r the distance to the center of the planet
we substitute
Since the speed is constant, we can use the uniform motion ratio
v =
In the case of a complete orbit, the time is called the period.
The distance traveled is the length of the orbit circle
Δx = 2π r
We substitute
Let's write this expression for each satellite
Io satellite
Let's call the distance from Jupiter is
r =
TIo² = (4pi² / GM) rIo³
Europe satellite
Distance from Jupiter is
We calculate
In conclusion, using the universal gravitation law and Newton's second law, we find that the answer for the relationship of the relation periods of the satellites is:
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