neptune has a mass that is about 17 times the mass of earth. the distance between the sun and neptune is…

Explanation:

Step1: Recall gravitational - force formula

The gravitational force formula is $F = G\frac{Mm}{r^{2}}$, where $F$ is the gravitational force, $G$ is the gravitational constant, $M$ and $m$ are the masses of the two objects, and $r$ is the distance between them. Let $M_s$ be the mass of the Sun, $m_e$ be the mass of the Earth, $r_{se}$ be the distance between the Sun and the Earth, $m_n$ be the mass of Neptune, and $r_{sn}$ be the distance between the Sun and Neptune. We know that $m_n = 17m_e$ and $r_{sn}=30.1r_{se}$. The gravitational force between the Sun and the Earth is $F_{se}=G\frac{M_sm_e}{r_{se}^{2}} = 3.5\times 10^{28}\text{ N}$, and the gravitational force between the Sun and Neptune is $F_{sn}=G\frac{M_sm_n}{r_{sn}^{2}}$.

Step2: Substitute the relationships

Substitute $m_n = 17m_e$ and $r_{sn}=30.1r_{se}$ into the formula for $F_{sn}$: [ \begin{align*} F_{sn}&=G\frac{M_s\times17m_e}{(30.1r_{se})^{2}}\ &=G\frac{17M_sm_e}{30.1^{2}r_{se}^{2}}\ &=\frac{17}{30.1^{2}}\times G\frac{M_sm_e}{r_{se}^{2}} \end{align*} ] Since $G\frac{M_sm_e}{r_{se}^{2}}=F_{se} = 3.5\times 10^{28}\text{ N}$, we have $F_{sn}=\frac{17}{30.1^{2}}\times3.5\times 10^{28}\text{ N}$.

Step3: Calculate the value

First, calculate $\frac{17}{30.1^{2}}=\frac{17}{906.01}\approx0.01876$. Then, $F_{sn}=0.01876\times3.5\times 10^{28}\text{ N}$. $0.01876\times3.5 = 0.06566$. So, $F_{sn}=0.06566\times 10^{28}\text{ N}=6.566\times 10^{26}\text{ N}\approx6\times 10^{26}\text{ N}$.

Answer:

$6\times 10^{26}\text{ N}$