Answer :
To solve this problem, we will use Newton's Law of Gravitation formula:
[tex]\[ F = G \frac{m_1 \cdot m_2}{r^2} \][/tex]
Where:
- [tex]\( F \)[/tex] is the gravitational force between two masses,
- [tex]\( G \)[/tex] is the gravitational constant ([tex]\(6.674 \times 10^{-11} \, \text{N m}^2/\text{kg}^2\)[/tex]),
- [tex]\( m_1 \)[/tex] and [tex]\( m_2 \)[/tex] are the masses,
- [tex]\( r \)[/tex] is the distance between the centers of the two masses.
Given two scenarios where the distance between the objects is different, we need to determine which configuration results in a weaker gravitational force.
### Scenario A: Distance is 3D
In this scenario, the objects are 3D apart. Substituting 3D for [tex]\( r \)[/tex] in the gravitational force formula, we get:
[tex]\[ F_A = G \frac{m_1 \cdot m_2}{(3D)^2} = G \frac{m_1 \cdot m_2}{9D^2} \][/tex]
### Scenario B: Distance is 1D
For the second scenario, the objects are 1D apart. Substituting 1D for [tex]\( r \)[/tex] in the gravitational force formula, we obtain:
[tex]\[ F_B = G \frac{m_1 \cdot m_2}{(1D)^2} = G \frac{m_1 \cdot m_2}{D^2} \][/tex]
### Comparison
To determine which force is weaker, compare the two forces directly.
[tex]\[ \frac{F_A}{F_B} = \frac{G \frac{m_1 \cdot m_2}{9D^2}}{G \frac{m_1 \cdot m_2}{D^2}} = \frac{1}{9} \][/tex]
This ratio tells us that [tex]\( F_A \)[/tex] is [tex]\(\frac{1}{9}\)[/tex] of [tex]\( F_B \)[/tex]. In other words, the force when the distance is 3D is nine times weaker than the force when the distance is 1D.
### Conclusion
The objects that are a distance of 3D apart will experience a weaker gravitational force compared to the objects that are only a distance of 1D apart. Thus, the correct answer is:
A) Objects are a distance of 3D apart.
[tex]\[ F = G \frac{m_1 \cdot m_2}{r^2} \][/tex]
Where:
- [tex]\( F \)[/tex] is the gravitational force between two masses,
- [tex]\( G \)[/tex] is the gravitational constant ([tex]\(6.674 \times 10^{-11} \, \text{N m}^2/\text{kg}^2\)[/tex]),
- [tex]\( m_1 \)[/tex] and [tex]\( m_2 \)[/tex] are the masses,
- [tex]\( r \)[/tex] is the distance between the centers of the two masses.
Given two scenarios where the distance between the objects is different, we need to determine which configuration results in a weaker gravitational force.
### Scenario A: Distance is 3D
In this scenario, the objects are 3D apart. Substituting 3D for [tex]\( r \)[/tex] in the gravitational force formula, we get:
[tex]\[ F_A = G \frac{m_1 \cdot m_2}{(3D)^2} = G \frac{m_1 \cdot m_2}{9D^2} \][/tex]
### Scenario B: Distance is 1D
For the second scenario, the objects are 1D apart. Substituting 1D for [tex]\( r \)[/tex] in the gravitational force formula, we obtain:
[tex]\[ F_B = G \frac{m_1 \cdot m_2}{(1D)^2} = G \frac{m_1 \cdot m_2}{D^2} \][/tex]
### Comparison
To determine which force is weaker, compare the two forces directly.
[tex]\[ \frac{F_A}{F_B} = \frac{G \frac{m_1 \cdot m_2}{9D^2}}{G \frac{m_1 \cdot m_2}{D^2}} = \frac{1}{9} \][/tex]
This ratio tells us that [tex]\( F_A \)[/tex] is [tex]\(\frac{1}{9}\)[/tex] of [tex]\( F_B \)[/tex]. In other words, the force when the distance is 3D is nine times weaker than the force when the distance is 1D.
### Conclusion
The objects that are a distance of 3D apart will experience a weaker gravitational force compared to the objects that are only a distance of 1D apart. Thus, the correct answer is:
A) Objects are a distance of 3D apart.
