To determine the gravitational force between Neptune and the Sun, given the mass of Neptune relative to Earth and the distance between Neptune and the Sun relative to the distance between Earth and the Sun, we can use the gravitational force formula and the provided ratios.
The gravitational force between two masses [tex]\( m_1 \)[/tex] and [tex]\( m_2 \)[/tex] separated by a distance [tex]\( r \)[/tex] is given by:
[tex]\[ F = G \frac{m_1 m_2}{r^2} \][/tex]
Where [tex]\( G \)[/tex] is the gravitational constant. For our purposes, we are comparing the gravitational forces in proportion by using the given relative values:
- The mass of Neptune is 17 times the mass of Earth.
- The distance between Neptune and the Sun is 30.1 times the distance between Earth and the Sun.
The gravitational force between the Sun and Earth is given as [tex]\( F_{Earth-Sun} = 3.5 \times 10^{28} \text{ N} \)[/tex]. To find the gravitational force between Neptune and the Sun ([tex]\( F_{Neptune-Sun} \)[/tex]), we use the ratios provided.
First, consider the proportional relationship of the forces. Since the gravitational force is proportional to the product of the masses and inversely proportional to the square of the distance:
[tex]\[ F_{Neptune-Sun} = F_{Earth-Sun} \left( \frac{m_{Neptune}}{m_{Earth}} \right) \left( \frac{d_{Earth-Sun}}{d_{Neptune-Sun}} \right)^2 \][/tex]
Given:
- [tex]\( \frac{m_{Neptune}}{m_{Earth}} = 17 \)[/tex]
- [tex]\( \frac{d_{Neptune-Sun}}{d_{Earth-Sun}} = 30.1 \)[/tex]
Rewriting the formula with these ratios:
[tex]\[ F_{Neptune-Sun} = 3.5 \times 10^{28} \text{ N} \left( \frac{17}{30.1^2} \right) \][/tex]
Calculate the exponent for the distance ratio:
[tex]\[ 30.1^2 = 906.01 \][/tex]
Now divide the mass ratio by the squared distance ratio:
[tex]\[ \frac{17}{906.01} \approx 0.01876 \][/tex]
Finally, calculate the force:
[tex]\[ F_{Neptune-Sun} = 3.5 \times 10^{28} \times 0.01876 \approx 6.567 \times 10^{26} \text{ N} \][/tex]
Thus, the gravitational force between Neptune and the Sun is approximately [tex]\( 6.567 \times 10^{26} \text{ N} \)[/tex].
Rounding this to the nearest power of ten in scientific notation, the force is closest to:
[tex]\[ 6.6 \times 10^{26} \text{ N} \][/tex]
Given the options:
- [tex]\( 6 \times 10^{25} \text{ N} \)[/tex]
- [tex]\( 6 \times 10^{27} \text{ N} \)[/tex]
- [tex]\( 6 \times 10^{24} \text{ N} \)[/tex]
- [tex]\( 6 \times 10^{29} \text{ N} \)[/tex]
The correct and closest option is:
[tex]\[ 6 \times 10^{27} \][/tex]
So, the closest answer is:
[tex]\[ 6 \times 10^{27} \text{ N} \][/tex]
