Chapter 7, Problem 47A

To determine

The rank pair of planets according to the gravitational force from least to greatest.

Answer to Problem 47A

The rank of the pairs of planets according to the gravitational force from least to greatest as,

D) the gravitational force for Mercury and Earth is $F=1.55\times {10}^{16}\text{ N}$ .

A) the gravitational force for Mercury and Venus is $F=4.29\times {10}^{16}\text{ N}$ .

E) the gravitational force for Jupiter and Mercury is $F=8.07\times {10}^{16}\text{ N}$ .

C) the gravitational force for Jupiter and Earth is $F=1.9\times {10}^{18}\text{ N}$ .

B) the gravitational force for Jupiter and Saturn is $F=1.65\times {10}^{20}\text{ N}$ .

Explanation of Solution

Given:

The distance between the centers of Mercury and Venus is $5.0\times {10}^7\text{ km}$ .

The distance between the centers of Jupiter and Saturn is $6.6\times {10}^8\text{ km}$ .

The distance between the centers of Jupiter and Earth is $6.3\times {10}^8\text{ km}$ .

The distance between the centers of Mercury and Earth is $9.2\times {10}^7\text{ km}$ .

The distance between the centers of Jupiter and Mercury is $7.2\times {10}^8\text{ km}$ .

Formula used:

The expression for the gravitational force is,

  $F=G\frac{m_1{m}_2}{r^2}$

Here, $G$ is the gravitational constant, $m_1$ is the mass of object 1, $m_2$ is the mass of object 2, and $r$ is the distance between the centers of the object.

Calculation:

Consider the value of gravitational constant $G=6.67\times {10}^{-11}\text{N}\cdot {\text{m}}^{\text{2}}/{\text{kg}}^{\text{2}}$ .

The mass of the Mercury is $m_{\text{M}}=0.3302\times {10}^{24}\text{ kg}$ .

The mass of the Venus is $m_{\text{V}}=4.8685\times {10}^{24}\text{ kg}$ .

The mass of the Jupiter is $m_{\text{J}}=1898.6\times {10}^{24}\text{ kg}$ .

The mass of the Saturn is $m_{\text{S}}=568.46\times {10}^{24}\text{ kg}$ .

The mass of the Earth is $m_{\text{E}}=5.9736\times {10}^{24}\text{ kg}$ .

(A)

Find the gravitational force for Mercury and Venus as follows:

  $\begin{array}{l}F=G\frac{m_{\text{M}}{m}_{\text{V}}}{r^2}\\ F=\left(6.67\times {10}^{-11}\text{N}\cdot {\text{m}}^{\text{2}}/{\text{kg}}^{\text{2}}\right)\times \frac{\left(0.3302\times {10}^{24}\text{ kg}\right)\times \left(4.8685\times {10}^{24}\text{ kg}\right)}{{\left(5.0\times {10}^7\text{ km}\times \frac{1,000\text{ m}}{1\text{ km}}\right)}^2}\\ F=4.29\times {10}^{16}\text{ N}\end{array}$

Hence, the gravitational force for Mercury and Venus is $F=4.29\times {10}^{16}\text{ N}$ .

(B)

Find the gravitational force for Jupiter and Saturn as follows:

  $\begin{array}{l}F=G\frac{m_{\text{J}}{m}_{\text{S}}}{r^2}\\ F=\left(6.67\times {10}^{-11}\text{N}\cdot {\text{m}}^{\text{2}}/{\text{kg}}^{\text{2}}\right)\times \frac{\left(1898.6\times {10}^{24}\text{ kg}\right)\times \left(568.46\times {10}^{24}\text{ kg}\right)}{{\left(6.6\times {10}^8\text{ km}\times \frac{1,000\text{ m}}{1\text{ km}}\right)}^2}\\ F=1.65\times {10}^{20}\text{ N}\end{array}$

Hence, the gravitational force for Jupiter and Saturn is $F=1.65\times {10}^{20}\text{ N}$ .

(C)

Find the gravitational force for Jupiter and Earth as follows:

  $\begin{array}{l}F=G\frac{m_{\text{J}}{m}_{\text{E}}}{r^2}\\ F=\left(6.67\times {10}^{-11}\text{N}\cdot {\text{m}}^{\text{2}}/{\text{kg}}^{\text{2}}\right)\times \frac{\left(1898.6\times {10}^{24}\text{ kg}\right)\times \left(5.9736\times {10}^{24}\text{ kg}\right)}{{\left(6.3\times {10}^8\text{ km}\times \frac{1,000\text{ m}}{1\text{ km}}\right)}^2}\\ F=1.9\times {10}^{18}\text{ N}\end{array}$

Hence, the gravitational force for Jupiter and Earth is $F=1.9\times {10}^{18}\text{ N}$ .

(D)

Find the gravitational force for Mercury and Earth as follows:

  $\begin{array}{l}F=G\frac{m_{\text{M}}{m}_{\text{E}}}{r^2}\\ F=\left(6.67\times {10}^{-11}\text{N}\cdot {\text{m}}^{\text{2}}/{\text{kg}}^{\text{2}}\right)\times \frac{\left(0.3302\times {10}^{24}\text{ kg}\right)\times \left(5.9736\times {10}^{24}\text{ kg}\right)}{{\left(9.2\times {10}^7\text{ km}\times \frac{1,000\text{ m}}{1\text{ km}}\right)}^2}\\ F=1.55\times {10}^{16}\text{ N}\end{array}$

Hence, the gravitational force for Mercury and Earth is $F=1.55\times {10}^{16}\text{ N}$ .

(E)

Find the gravitational force for Jupiter and Mercury as follows:

  $\begin{array}{l}F=G\frac{m_{\text{J}}{m}_{\text{M}}}{r^2}\\ F=\left(6.67\times {10}^{-11}\text{N}\cdot {\text{m}}^{\text{2}}/{\text{kg}}^{\text{2}}\right)\times \frac{\left(1898.6\times {10}^{24}\text{ kg}\right)\times \left(0.3302\times {10}^{24}\text{ kg}\right)}{{\left(7.2\times {10}^8\text{ km}\times \frac{1,000\text{ m}}{1\text{ km}}\right)}^2}\\ F=8.07\times {10}^{16}\text{ N}\end{array}$

Hence, the gravitational force for Jupiter and Mercury is $F=8.07\times {10}^{16}\text{ N}$ .

Rank the pairs of planets according to the gravitational force from least to greatest as follows:

D) the gravitational force for Mercury and Earth is $F=1.55\times {10}^{16}\text{ N}$ .

A) the gravitational force for Mercury and Venus is $F=4.29\times {10}^{16}\text{ N}$ .

E) the gravitational force for Jupiter and Mercury is $F=8.07\times {10}^{16}\text{ N}$ .

C) the gravitational force for Jupiter and Earth is $F=1.9\times {10}^{18}\text{ N}$ .

B) the gravitational force for Jupiter and Saturn is $F=1.65\times {10}^{20}\text{ N}$ .

Conclusion:

Thus, the rank of the pairs of planets according to the gravitational force from least to greatest as,

D) the gravitational force for Mercury and Earth is $F=1.55\times {10}^{16}\text{ N}$ .

A) the gravitational force for Mercury and Venus is $F=4.29\times {10}^{16}\text{ N}$ .

E) the gravitational force for Jupiter and Mercury is $F=8.07\times {10}^{16}\text{ N}$ .

C) the gravitational force for Jupiter and Earth is $F=1.9\times {10}^{18}\text{ N}$ .

B) the gravitational force for Jupiter and Saturn is $F=1.65\times {10}^{20}\text{ N}$ .