Answer :
Sure! Let's break down the calculation step by step to determine the gravitational force between Neptune and the Sun based on the information provided.
### Step-by-Step Solution:
1. Identify the given data:
- Mass of Neptune ( [tex]\( m_{Neptune} \)[/tex] ) = 17 times the mass of Earth ( [tex]\( m_{Earth} \)[/tex] )
- Distance between the Sun and Neptune ( [tex]\( d_{Neptune} \)[/tex] ) = 30.1 times the distance between the Sun and Earth ( [tex]\( d_{Earth} \)[/tex] )
- Gravitational force between the Sun and Earth ( [tex]\( F_{Earth-Sun} \)[/tex] ) = [tex]\( 3.5 \times 10^{28} \)[/tex] N
2. State the gravitational force formula:
[tex]\[ F = G \frac{m_1 m_2}{r^2} \][/tex]
However, since we are considering ratios, the gravitational constant [tex]\( G \)[/tex] cancels out.
3. Calculate the ratio of the forces:
Given that the mass of Neptune is 17 times the mass of Earth, we have:
[tex]\[ \frac{F_{Neptune-Sun}}{F_{Earth-Sun}} = \frac{m_{Neptune} / m_{Earth}}{(d_{Neptune} / d_{Earth})^2} \][/tex]
4. Substitute the given ratios:
- Mass ratio [tex]\( = 17 \)[/tex]
- Distance ratio [tex]\( = 30.1 \)[/tex]
Plugging in these values, we get:
[tex]\[ \frac{F_{Neptune-Sun}}{F_{Earth-Sun}} = \frac{17}{(30.1)^2} \][/tex]
5. Calculate the distance squared term:
[tex]\[ (30.1)^2 = 906.01 \][/tex]
6. Calculate the force ratio:
[tex]\[ \frac{F_{Neptune-Sun}}{F_{Earth-Sun}} = \frac{17}{906.01} \approx 0.01876 \][/tex]
7. Calculate the actual force between Neptune and the Sun:
Using the force ratio, multiply by the gravitational force between the Sun and Earth:
[tex]\[ F_{Neptune-Sun} = F_{Earth-Sun} \times 0.01876 \][/tex]
[tex]\[ F_{Neptune-Sun} = 3.5 \times 10^{28} \times 0.01876 \][/tex]
[tex]\[ F_{Neptune-Sun} \approx 6.567256432048211 \times 10^{26} \][/tex]
### Conclusion:
We have determined that the gravitational force between the Sun and Neptune is approximately [tex]\( 6.567256432048211 \times 10^{26} \)[/tex] N. Among the given options, the closest value to this result is:
[tex]\[ 6 \times 10^{26} \][/tex] N.
Therefore, the correct answer is:
[tex]\[ \boxed{6 \times 10^{26} \text{ N}} \][/tex]
### Step-by-Step Solution:
1. Identify the given data:
- Mass of Neptune ( [tex]\( m_{Neptune} \)[/tex] ) = 17 times the mass of Earth ( [tex]\( m_{Earth} \)[/tex] )
- Distance between the Sun and Neptune ( [tex]\( d_{Neptune} \)[/tex] ) = 30.1 times the distance between the Sun and Earth ( [tex]\( d_{Earth} \)[/tex] )
- Gravitational force between the Sun and Earth ( [tex]\( F_{Earth-Sun} \)[/tex] ) = [tex]\( 3.5 \times 10^{28} \)[/tex] N
2. State the gravitational force formula:
[tex]\[ F = G \frac{m_1 m_2}{r^2} \][/tex]
However, since we are considering ratios, the gravitational constant [tex]\( G \)[/tex] cancels out.
3. Calculate the ratio of the forces:
Given that the mass of Neptune is 17 times the mass of Earth, we have:
[tex]\[ \frac{F_{Neptune-Sun}}{F_{Earth-Sun}} = \frac{m_{Neptune} / m_{Earth}}{(d_{Neptune} / d_{Earth})^2} \][/tex]
4. Substitute the given ratios:
- Mass ratio [tex]\( = 17 \)[/tex]
- Distance ratio [tex]\( = 30.1 \)[/tex]
Plugging in these values, we get:
[tex]\[ \frac{F_{Neptune-Sun}}{F_{Earth-Sun}} = \frac{17}{(30.1)^2} \][/tex]
5. Calculate the distance squared term:
[tex]\[ (30.1)^2 = 906.01 \][/tex]
6. Calculate the force ratio:
[tex]\[ \frac{F_{Neptune-Sun}}{F_{Earth-Sun}} = \frac{17}{906.01} \approx 0.01876 \][/tex]
7. Calculate the actual force between Neptune and the Sun:
Using the force ratio, multiply by the gravitational force between the Sun and Earth:
[tex]\[ F_{Neptune-Sun} = F_{Earth-Sun} \times 0.01876 \][/tex]
[tex]\[ F_{Neptune-Sun} = 3.5 \times 10^{28} \times 0.01876 \][/tex]
[tex]\[ F_{Neptune-Sun} \approx 6.567256432048211 \times 10^{26} \][/tex]
### Conclusion:
We have determined that the gravitational force between the Sun and Neptune is approximately [tex]\( 6.567256432048211 \times 10^{26} \)[/tex] N. Among the given options, the closest value to this result is:
[tex]\[ 6 \times 10^{26} \][/tex] N.
Therefore, the correct answer is:
[tex]\[ \boxed{6 \times 10^{26} \text{ N}} \][/tex]
