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Neptune has a mass that is about 17 times the mass of Earth. The distance between the Sun and Neptune is about 30.1 times the distance between the Sun and Earth.

If the gravitational force between the Sun and Earth is [tex][tex]$3.5 \times 10^{28} \, \text{N}$[/tex][/tex], which is closest to the force between Neptune and the Sun?

A. [tex][tex]$6 \times 10^{26} \, \text{N}$[/tex][/tex]
B. [tex][tex]$6 \times 10^{27} \, \text{N}$[/tex][/tex]
C. [tex][tex]$6 \times 10^{28} \, \text{N}$[/tex][/tex]
D. [tex][tex]$6 \times 10^{29} \, \text{N}$[/tex][/tex]

Sagot :

GinnyAnswer
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]
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