To determine the new gravitational force between the Earth and the satellite when the distance is increased by a factor of 4, follow these steps:
1. Understand the gravitational force law:
The gravitational force between two masses is inversely proportional to the square of the distance between them. This means if the distance is [tex]\( d \)[/tex], the gravitational force [tex]\( F \)[/tex] is proportional to [tex]\( \frac{1}{d^2} \)[/tex].
2. Original force and distance information:
The original gravitational force between the Earth and the satellite is 243 N, at some initial distance [tex]\( d \)[/tex].
3. Effect of increasing the distance by a factor of 4:
If the distance increases to 4 times the original distance (i.e., [tex]\( 4d \)[/tex]), then according to the inverse square law, the force will be reduced by a factor of [tex]\( (4)^2 \)[/tex].
4. Calculate the factor reduction:
[tex]\[
\text{Reduction factor} = 4^2 = 16
\][/tex]
5. Determine the new force:
The new force is obtained by dividing the original force by this factor:
[tex]\[
\text{New Force} = \frac{\text{Original Force}}{\text{Reduction Factor}} = \frac{243}{16}
\][/tex]
6. Perform the division to find the precise force:
[tex]\[
\text{New Force} = \frac{243}{16} \approx 15.1875
\][/tex]
7. Round the result to the nearest whole number:
We round 15.1875 to the nearest whole number, which gives 15.
Conclusion:
The new gravitational force between the Earth and the satellite, when it is moved to 4 times the original distance, is approximately [tex]\( 15 \)[/tex] N.
