To determine the new gravitational force between two objects when the distance between them doubles, we need to understand how the gravitational force is affected by distance. According to Newton's law of universal gravitation, the gravitational force ([tex]\(F\)[/tex]) between two masses is inversely proportional to the square of the distance ([tex]\(d\)[/tex]) between them.
The formula for the gravitational force between two objects is:
[tex]\[ F \propto \frac{1}{d^2} \][/tex]
Given:
- Initial gravitational force ([tex]\(F_{\text{initial}}\)[/tex]) = 1600 N
- Initial distance between the objects = [tex]\(d\)[/tex]
- New distance between the objects = [tex]\(2d\)[/tex] (since the distance is doubled)
Since the gravitational force is inversely proportional to the square of the distance:
[tex]\[ F_{\text{new}} \propto \frac{1}{(2d)^2} \][/tex]
[tex]\[ F_{\text{new}} \propto \frac{1}{4d^2} \][/tex]
Now, we know that initially:
[tex]\[ F_{\text{initial}} \propto \frac{1}{d^2} \][/tex]
To find the new gravitational force ([tex]\(F_{\text{new}}\)[/tex]), we can set up the proportion:
[tex]\[ \frac{F_{\text{new}}}{F_{\text{initial}}} = \frac{\frac{1}{4d^2}}{\frac{1}{d^2}} \][/tex]
Simplify the right side of the equation:
[tex]\[ \frac{F_{\text{new}}}{F_{\text{initial}}} = \frac{1}{4} \][/tex]
Thus:
[tex]\[ F_{\text{new}} = \frac{F_{\text{initial}}}{4} \][/tex]
Substitute the initial force:
[tex]\[ F_{\text{new}} = \frac{1600 \, \text{N}}{4} \][/tex]
Perform the division:
[tex]\[ F_{\text{new}} = 400 \, \text{N} \][/tex]
Therefore, the new gravitational force between the objects, when the distance between them doubles, is [tex]\(400 \, \text{N}\)[/tex]. The correct option is:
[tex]\[ 400 \, \text{N} \][/tex]
