To solve this problem, we need to understand the relationship between distance and the force of attraction between two objects. Specifically, we are looking at the inverse square law, which states that the force of attraction \( F \) is inversely proportional to the square of the distance \( d \) between the objects. Mathematically, this relationship can be expressed as:
[tex]\[ F \propto \frac{1}{d^2} \][/tex]
Now, let's break down the steps to determine how increasing the distance affects the force of attraction.
1. Original Parameters:
- Let's assume the initial distance between the two objects is \( d \).
- The initial force of attraction between the two objects is \( F \).
2. New Distance:
- The distance between the objects is increased by 3 times the original distance. So, the new distance is \( 3d \).
3. Calculating the New Force:
- According to the inverse square law, the new force \( F_{\text{new}} \) can be calculated as follows:
[tex]\[ F_{\text{new}} \propto \frac{1}{(3d)^2} \][/tex]
Simplifying this, we get:
[tex]\[ F_{\text{new}} \propto \frac{1}{9d^2} \][/tex]
Since the original force \( F \propto \frac{1}{d^2} \), we can compare the original force to the new force.
4. Proportionality Factor:
- We need to find the proportionality factor between \( F_{\text{new}} \) and \( F \). By substituting the proportionality constants, we get:
[tex]\[ F_{\text{new}} = F \cdot \frac{1}{9} \][/tex]
Therefore, the new force is:
[tex]\[ F_{\text{new}} = \frac{F}{9} \][/tex]
This shows that the new force of attraction is \(\frac{1}{9}\) of the original force.
Given the options:
A. 1 - The new force will be \(\frac{1}{3}\) of the original.
B. The new force will be 3 times more than the original.
C. The new force will be \(\frac{1}{9}\) of the original.
D. The new force will be 9 times more than the original.
Thus, the correct answer is:
C. The new force will be [tex]\(\frac{1}{9}\)[/tex] of the original.
