Sure, let's go through the detailed step-by-step process to determine how the force of attraction between two objects changes when the distance between them is increased by a factor of 3.
### Step-by-Step Solution:
1. Understanding the Law:
According to Newton's Law of Universal Gravitation, the force of attraction [tex]\( F \)[/tex] between two objects is given by:
[tex]\[
F = \frac{G \cdot m_1 \cdot m_2}{r^2}
\][/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.
- [tex]\( r \)[/tex] is the distance between the centers of the two objects.
2. Initial Force Calculation:
Let's denote the initial force of attraction as [tex]\( F_0 \)[/tex] when the distance is [tex]\( r \)[/tex].
[tex]\[
F_0 = \frac{G \cdot m_1 \cdot m_2}{r^2}
\][/tex]
3. Increasing the Distance:
Now, the distance [tex]\( r \)[/tex] is increased by a factor of 3. The new distance [tex]\( r' \)[/tex] will be:
[tex]\[
r' = 3r
\][/tex]
4. Calculating the New Force:
With the new distance [tex]\( r' \)[/tex], the new force of attraction [tex]\( F' \)[/tex] will be:
[tex]\[
F' = \frac{G \cdot m_1 \cdot m_2}{(r')^2}
\][/tex]
Substituting [tex]\( r' = 3r \)[/tex]:
[tex]\[
F' = \frac{G \cdot m_1 \cdot m_2}{(3r)^2}
\][/tex]
[tex]\[
F' = \frac{G \cdot m_1 \cdot m_2}{9r^2}
\][/tex]
5. Finding the Factor:
Next, we compare [tex]\( F' \)[/tex] with the initial force [tex]\( F_0 \)[/tex]:
[tex]\[
F_0 = \frac{G \cdot m_1 \cdot m_2}{r^2}
\][/tex]
[tex]\[
F' = \frac{G \cdot m_1 \cdot m_2}{9r^2} = \frac{F_0}{9}
\][/tex]
So, the new force of attraction [tex]\( F' \)[/tex] is [tex]\(\frac{1}{9}\)[/tex] of the original force [tex]\( F_0 \)[/tex].
### Conclusion:
When the distance between two objects is increased by a factor of 3, the gravitational force of attraction between them is reduced to [tex]\(\frac{1}{9}\)[/tex] of the original force.
Therefore, the correct answer is:
C. The new force will be [tex]\(\frac{1}{9}\)[/tex] of the original.
