Sure, let's break down the problem step-by-step:
The gravitational force of attraction between two objects is governed by Newton's Law of Gravitation, which is given by the formula:
[tex]\[ F = \frac{G m_1 m_2}{r^2} \][/tex]
Where:
- [tex]\( F \)[/tex] is the force of attraction between the two objects.
- [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.
According to the question, the distance between the two objects is increased by a factor of 3 times the original distance. Let [tex]\( r_{new} \)[/tex] be the new distance, then:
[tex]\[ r_{new} = 3r \][/tex]
We need to find the new force of attraction [tex]\( F_{new} \)[/tex], when the distance is [tex]\( 3r \)[/tex].
Substitute [tex]\( r_{new} \)[/tex] into the original formula:
[tex]\[ F_{new} = \frac{G m_1 m_2}{(3r)^2} \][/tex]
[tex]\[ F_{new} = \frac{G m_1 m_2}{9r^2} \][/tex]
From the original force equation, we know:
[tex]\[ F = \frac{G m_1 m_2}{r^2} \][/tex]
Therefore, the new force [tex]\( F_{new} \)[/tex] can be written as:
[tex]\[ F_{new} = \frac{F}{9} \][/tex]
This means that the new force is one-ninth ( [tex]\(\frac{1}{9} \)[/tex] ) of the original force.
So, the correct answer to the given question is:
C. The new force will be [tex]\(\frac{1}{9}\)[/tex] of the original.
