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Sagot :
To determine the scenario with the least gravitational force between two objects, we will utilize the formula for gravitational force:
[tex]\[ F = G \frac{m_1 m_2}{r^2} \][/tex]
where [tex]\( G \)[/tex] is the gravitational constant, [tex]\( m_1 \)[/tex] and [tex]\( m_2 \)[/tex] are the masses of the two objects, and [tex]\( r \)[/tex] is the distance between the centers of the two masses.
Let's go through each scenario step-by-step:
### Scenario A:
- Mass of object 1 ([tex]\( m_1 \)[/tex]): 12 kg
- Mass of object 2 ([tex]\( m_2 \)[/tex]): 12 kg
- Distance ([tex]\( r \)[/tex]): 1.5 m
Gravitational force equation:
[tex]\[ F_A = G \frac{12 \times 12}{(1.5)^2} \][/tex]
[tex]\[ F_A = G \frac{144}{2.25} \][/tex]
[tex]\[ F_A = 64 G \][/tex]
Numerically, this gives:
[tex]\[ F_A = 4.271552 \times 10^{-9} \, \text{N} \][/tex]
### Scenario B:
- Mass of object 1 ([tex]\( m_1 \)[/tex]): 15 kg
- Mass of object 2 ([tex]\( m_2 \)[/tex]): 12 kg
- Distance ([tex]\( r \)[/tex]): 1.5 m
Gravitational force equation:
[tex]\[ F_B = G \frac{15 \times 12}{(1.5)^2} \][/tex]
[tex]\[ F_B = G \frac{180}{2.25} \][/tex]
[tex]\[ F_B = 80 G \][/tex]
Numerically, this gives:
[tex]\[ F_B = 5.339440 \times 10^{-9} \, \text{N} \][/tex]
### Scenario C:
- Mass of object 1 ([tex]\( m_1 \)[/tex]): 15 kg
- Mass of object 2 ([tex]\( m_2 \)[/tex]): 12 kg
- Distance ([tex]\( r \)[/tex]): 0.5 m
Gravitational force equation:
[tex]\[ F_C = G \frac{15 \times 12}{(0.5)^2} \][/tex]
[tex]\[ F_C = G \frac{180}{0.25} \][/tex]
[tex]\[ F_C = 720 G \][/tex]
Numerically, this gives:
[tex]\[ F_C = 48.05496 \times 10^{-9} \, \text{N} \][/tex]
### Scenario D:
- Mass of object 1 ([tex]\( m_1 \)[/tex]): 12 kg
- Mass of object 2 ([tex]\( m_2 \)[/tex]): 12 kg
- Distance ([tex]\( r \)[/tex]): 0.5 m
Gravitational force equation:
[tex]\[ F_D = G \frac{12 \times 12}{(0.5)^2} \][/tex]
[tex]\[ F_D = G \frac{144}{0.25} \][/tex]
[tex]\[ F_D = 576 G \][/tex]
Numerically, this gives:
[tex]\[ F_D = 38.443968 \times 10^{-9} \, \text{N} \][/tex]
### Conclusion:
Comparing the calculated forces:
- [tex]\( F_A = 4.271552 \times 10^{-9} \, \text{N} \)[/tex]
- [tex]\( F_B = 5.339440 \times 10^{-9} \, \text{N} \)[/tex]
- [tex]\( F_C = 48.05496 \times 10^{-9} \, \text{N} \)[/tex]
- [tex]\( F_D = 38.443968 \times 10^{-9} \, \text{N} \)[/tex]
The smallest force is [tex]\( F_A = 4.271552 \times 10^{-9} \, \text{N} \)[/tex], which occurs in Scenario A.
Thus, the scenario with the least gravitational force between the objects is Scenario A.
[tex]\[ F = G \frac{m_1 m_2}{r^2} \][/tex]
where [tex]\( G \)[/tex] is the gravitational constant, [tex]\( m_1 \)[/tex] and [tex]\( m_2 \)[/tex] are the masses of the two objects, and [tex]\( r \)[/tex] is the distance between the centers of the two masses.
Let's go through each scenario step-by-step:
### Scenario A:
- Mass of object 1 ([tex]\( m_1 \)[/tex]): 12 kg
- Mass of object 2 ([tex]\( m_2 \)[/tex]): 12 kg
- Distance ([tex]\( r \)[/tex]): 1.5 m
Gravitational force equation:
[tex]\[ F_A = G \frac{12 \times 12}{(1.5)^2} \][/tex]
[tex]\[ F_A = G \frac{144}{2.25} \][/tex]
[tex]\[ F_A = 64 G \][/tex]
Numerically, this gives:
[tex]\[ F_A = 4.271552 \times 10^{-9} \, \text{N} \][/tex]
### Scenario B:
- Mass of object 1 ([tex]\( m_1 \)[/tex]): 15 kg
- Mass of object 2 ([tex]\( m_2 \)[/tex]): 12 kg
- Distance ([tex]\( r \)[/tex]): 1.5 m
Gravitational force equation:
[tex]\[ F_B = G \frac{15 \times 12}{(1.5)^2} \][/tex]
[tex]\[ F_B = G \frac{180}{2.25} \][/tex]
[tex]\[ F_B = 80 G \][/tex]
Numerically, this gives:
[tex]\[ F_B = 5.339440 \times 10^{-9} \, \text{N} \][/tex]
### Scenario C:
- Mass of object 1 ([tex]\( m_1 \)[/tex]): 15 kg
- Mass of object 2 ([tex]\( m_2 \)[/tex]): 12 kg
- Distance ([tex]\( r \)[/tex]): 0.5 m
Gravitational force equation:
[tex]\[ F_C = G \frac{15 \times 12}{(0.5)^2} \][/tex]
[tex]\[ F_C = G \frac{180}{0.25} \][/tex]
[tex]\[ F_C = 720 G \][/tex]
Numerically, this gives:
[tex]\[ F_C = 48.05496 \times 10^{-9} \, \text{N} \][/tex]
### Scenario D:
- Mass of object 1 ([tex]\( m_1 \)[/tex]): 12 kg
- Mass of object 2 ([tex]\( m_2 \)[/tex]): 12 kg
- Distance ([tex]\( r \)[/tex]): 0.5 m
Gravitational force equation:
[tex]\[ F_D = G \frac{12 \times 12}{(0.5)^2} \][/tex]
[tex]\[ F_D = G \frac{144}{0.25} \][/tex]
[tex]\[ F_D = 576 G \][/tex]
Numerically, this gives:
[tex]\[ F_D = 38.443968 \times 10^{-9} \, \text{N} \][/tex]
### Conclusion:
Comparing the calculated forces:
- [tex]\( F_A = 4.271552 \times 10^{-9} \, \text{N} \)[/tex]
- [tex]\( F_B = 5.339440 \times 10^{-9} \, \text{N} \)[/tex]
- [tex]\( F_C = 48.05496 \times 10^{-9} \, \text{N} \)[/tex]
- [tex]\( F_D = 38.443968 \times 10^{-9} \, \text{N} \)[/tex]
The smallest force is [tex]\( F_A = 4.271552 \times 10^{-9} \, \text{N} \)[/tex], which occurs in Scenario A.
Thus, the scenario with the least gravitational force between the objects is Scenario A.
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