To determine the gravitational force between two bodies when the distance between them is changed, we need to understand the inverse square law of gravitation. According to this law, the gravitational force [tex]\( F \)[/tex] between two bodies is inversely proportional to the square of the distance [tex]\( r \)[/tex] between their centers. This relationship can be expressed mathematically as:
[tex]\[ F \propto \frac{1}{r^2} \][/tex]
Given:
- Initial distance ([tex]\( r_1 \)[/tex]) between the centers of the bodies = 2 m
- Initial gravitational force ([tex]\( F_1 \)[/tex]) = 200 N
- New distance ([tex]\( r_2 \)[/tex]) between the centers of the bodies = 4 m
We need to find the new gravitational force ([tex]\( F_2 \)[/tex]) when the distance between the bodies is increased to 4 m.
According to the inverse square law:
[tex]\[ \frac{F_2}{F_1} = \left( \frac{r_1}{r_2} \right)^2 \][/tex]
By substituting the known values, we can solve for [tex]\( F_2 \)[/tex]:
[tex]\[ \frac{F_2}{200} = \left( \frac{2}{4} \right)^2 \][/tex]
[tex]\[ \frac{F_2}{200} = \left( \frac{1}{2} \right)^2 \][/tex]
[tex]\[ \frac{F_2}{200} = \frac{1}{4} \][/tex]
Multiplying both sides by 200 to isolate [tex]\( F_2 \)[/tex]:
[tex]\[ F_2 = 200 \times \frac{1}{4} \][/tex]
[tex]\[ F_2 = 50 \][/tex]
Therefore, the new gravitational force between the two bodies, when the distance between them is increased to 4 meters, is 50 N.
