Alright, let's solve the problem step by step using the principles of gravitational force.
### Gravitational Force Formula:
The gravitational force [tex]\( F \)[/tex] between two masses is given by Newton's Law of Universal Gravitation, which states:
[tex]\[ F = G \frac{M_{1} M_{2}}{d^2} \][/tex]
Where:
- [tex]\( F \)[/tex] is the gravitational force between the masses.
- [tex]\( G \)[/tex] is the gravitational constant, [tex]\(6.67430 \times 10^{-11} \, \text{m}^3 \text{kg}^{-1} \text{s}^{-2}\)[/tex].
- [tex]\( M_{1} \)[/tex] and [tex]\( M_{2} \)[/tex] are the masses (in this case, [tex]\( M_{1} \)[/tex] is the mass of the moon and [tex]\( M_{2} \)[/tex] is the mass of the satellite).
- [tex]\( d \)[/tex] is the distance between the centers of the two masses.
### Given Data:
- Gravitational force [tex]\( F = 324 \, \text{N} \)[/tex]
- Mass of the moon [tex]\( M_{moon} = 7.3 \times 10^{22} \, \text{kg} \)[/tex]
- Distance [tex]\( d = 2.6 \times 10^6 \, \text{m} \)[/tex]
### Step-by-Step Solution:
1. Rearrange the formula to solve for [tex]\( M_{2} \)[/tex] (mass of the satellite):
[tex]\[ M_{2} = \frac{F d^2}{G M_{1}} \][/tex]
2. Substitute the given values into the equation:
[tex]\[ M_{satellite} = \frac{324 \, \text{N} \times (2.6 \times 10^6 \, \text{m})^2}{6.67430 \times 10^{-11} \, \text{m}^3 \text{kg}^{-1} \text{s}^{-2} \times 7.3 \times 10^{22} \, \text{kg}} \][/tex]
3. Calculate each part:
- [tex]\( d^2 = (2.6 \times 10^6 \, \text{m})^2 = 6.76 \times 10^{12} \, \text{m}^2 \)[/tex]
- [tex]\( F d^2 = 324 \, \text{N} \times 6.76 \times 10^{12} \, \text{m}^2 = 2.191824 \times 10^{15} \, \text{N} \, \text{m}^2 \)[/tex]
4. Calculate the denominator:
[tex]\[ G M_{moon} = 6.67430 \times 10^{-11} \, \text{m}^3 \text{kg}^{-1} \text{s}^{-2} \times 7.3 \times 10^{22} \, \text{kg} = 4.871239 \times 10^{12} \, \text{N} \, \text{m}^2 \][/tex]
5. Calculate the mass of the satellite:
[tex]\[ M_{satellite} = \frac{2.191824 \times 10^{15} \, \text{N} \text{m}^2}{4.871239 \times 10^{12} \, \text{N} \text{m}^2 \text{kg}^{-1}} = 449.5345979538361 \, \text{kg} \][/tex]
### Conclusion:
The mass of the satellite is approximately 450 kg.
The closest answer choice to this calculation is:
[tex]\[ \boxed{450 \, \text{kg}} \][/tex]
