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Sagot :
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.
### 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.
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