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:
[tex]\frac{T_{Eu}}{T_{Io}}[/tex] = 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
[tex]F = G \frac{Mm}{r^2}[/tex]
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 are the satellites of Jupiter planet,
[tex]G \frac{Mm}{r^2} = m a[/tex]
Suppose the motion of the satellites is circular, then the acceleration is centripetal
a = [tex]\frac{v^2}{r}[/tex]
Where v is the speed of the satellite and r the distance to the center of the planet
we substitute
[tex]G \frac{Mm}{r^2} = m \frac{v^2}{r} \\G \frac{M}{r} = v^2[/tex]
Since the speed is constant, we can use the uniform motion ratio
v = [tex]\frac{\Delta x}{t}[/tex]
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
[tex]G \frac{M}{r} = (\frac{2 \pi r}{T} )^2 \\T^2 = (\frac{4 \pi ^2}{GM}) \ r^3[/tex]
Let's write this expression for each satellite
Io satellite
Let's call the distance from Jupiter is
r = [tex]r_{Io}[/tex]
Europe satellite
Distance from Jupiter is
[tex]r_{Eu} = 2 \ r_{Io}[/tex]
We calculate
[tex]T_{Eu} = ( \frac{4\pi ^2 }{GM} (2 \ r_{Io})^3\\T_{Eu} = ( \frac{4 \pi ^2 }{GM}) r_{Io} \ 8[/tex]
[tex]T_{Eu}^2 = 8 T_{Io}^2[/tex]
[tex]\frac{ T_{Eu}}{T_{Io}} = \sqrt{8} = 2.83[/tex]
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:
[tex]\frac{T_{Eu}}{T_{Io}} = 2.83[/tex]
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