Review the table detailing the mass of Jupiter's moons and their distance from the planet.

\begin{tabular}{|c|c|c|}
\hline Moon & Mass ( [tex]$\times 10^{19} kg$[/tex] ) & Distance from Jupiter [tex]$( km )$[/tex] \\
\hline Io & 8.932 & 421,700 \\
\hline Europa & 4.8 & 671,034 \\
\hline Ganymede & 14.819 & [tex]$1,070,412$[/tex] \\
\hline Callisto & 10.759 & [tex]$1,882,709$[/tex] \\
\hline
\end{tabular}

Which of Jupiter's moons has the greatest gravitational force with Jupiter?

A. Io
B. Europa
C. Ganymede
D. Callisto



Answer :

Let's determine which of Jupiter's moons has the greatest gravitational force with Jupiter by analyzing their gravitational forces.

### Given Data:

Moon Data:
- Io:
- Mass: [tex]\(8.932 \times 10^{19}\)[/tex] kg
- Distance from Jupiter: 421,700 km

- Europa:
- Mass: [tex]\(4.8 \times 10^{19}\)[/tex] kg
- Distance from Jupiter: 671,034 km

- Ganymede:
- Mass: [tex]\(14.819 \times 10^{19}\)[/tex] kg
- Distance from Jupiter: 1,070,412 km

- Callisto:
- Mass: [tex]\(10.759 \times 10^{19}\)[/tex] kg
- Distance from Jupiter: 1,882,709 km

Constants:
- Gravitational constant ([tex]\(G\)[/tex]): [tex]\(6.67430 \times 10^{-11}\)[/tex] m³ kg⁻¹ s⁻²
- Mass of Jupiter: [tex]\(1.898 \times 10^{27}\)[/tex] kg

### Conversion:
Convert distances from km to meters:

- Io: [tex]\(421,700 \times 10^3\)[/tex] m
- Europa: [tex]\(671,034 \times 10^3\)[/tex] m
- Ganymede: [tex]\(1,070,412 \times 10^3\)[/tex] m
- Callisto: [tex]\(1,882,709 \times 10^3\)[/tex] m

### Calculating Gravitational Force:

Using Newton's law of gravitation, which states the force between two masses is given by:
[tex]\[ F = G \frac{m_1 m_2}{r^2} \][/tex]

For Io:
[tex]\[ F_{\text{Io}} = 6.67430 \times 10^{-11} \frac{(1.898 \times 10^{27}) (8.932 \times 10^{19})}{(421,700 \times 10^3)^2} \approx 6.36272926176099 \times 10^{19} \, \text{N} \][/tex]

For Europa:
[tex]\[ F_{\text{Europa}} = 6.67430 \times 10^{-11} \frac{(1.898 \times 10^{27}) (4.8 \times 10^{19})}{(671,034 \times 10^3)^2} \approx 1.350374156878066 \times 10^{19} \, \text{N} \][/tex]

For Ganymede:
[tex]\[ F_{\text{Ganymede}} = 6.67430 \times 10^{-11} \frac{(1.898 \times 10^{27}) (14.819 \times 10^{19})}{(1,070,412 \times 10^3)^2} \approx 1.6383960469311234 \times 10^{19} \, \text{N} \][/tex]

For Callisto:
[tex]\[ F_{\text{Callisto}} = 6.67430 \times 10^{-11} \frac{(1.898 \times 10^{27}) (10.759 \times 10^{19})}{(1,882,709 \times 10^3)^2} \approx 3.8450982545003044 \times 10^{18} \, \text{N} \][/tex]

### Conclusion:

By comparing the gravitational forces:
- [tex]\( F_{\text{Io}} \approx 6.36272926176099 \times 10^{19} \, \text{N} \)[/tex]
- [tex]\( F_{\text{Europa}} \approx 1.350374156878066 \times 10^{19} \, \text{N} \)[/tex]
- [tex]\( F_{\text{Ganymede}} \approx 1.6383960469311234 \times 10^{19} \, \text{N} \)[/tex]
- [tex]\( F_{\text{Callisto}} \approx 3.8450982545003044 \times 10^{18} \, \text{N} \)[/tex]

The moon that experiences the greatest gravitational force with Jupiter is Io with a force of approximately [tex]\(6.36272926176099 \times 10^{19}\)[/tex] N.