Chapter 5, Problem 54P

To determine

The gravitational force of Jupiter on the earth, and compare the result with the magnitude of gravitational force on earth from the sun.

Answer to Problem 54P

The gravitational force of Jupiter on the earth when it is closest to earth is $\underset{\_}{1.9\times {10}^{18}\text{N}}$, The gravitational force of Jupiter on the earth when earth is farthest to earth is $\underset{\_}{8.8\times {10}^{17}\text{N}}$, and the gravitational force of the sun on the earth is $\underset{\_}{3.5\times {10}^{22}\text{N}}$.

Explanation of Solution

Write the expression for gravitational force of the sun on the earth.

    $F_{\text{SE}\text{grav}}=\frac{G{M}_{\text{Sun}}{M}_{\text{Earth}}}{r_{\text{Earth}}^2}$        (I)

Here, $F_{\text{SE}\text{grav}}$ is the gravitational force of the sun on the earth, $G$ is the universal gravitation constant, $M_{\text{Sun}}$ is the mass of the sun, $M_{\text{Earth}}$ is the mass of earth, and $r_{\text{Earth}}$ is the radius of earth.

Write the expression for gravitational force of Jupiter on the earth when it is closest to earth.

    $F_{\text{Jupiter closest }}=\frac{G{M}_{\text{Jupiter }}{M}_{\text{Earth }}}{{\left(r_{\text{Jupiter }}-r_{\text{Earth }}\right)}^2}$        (II)

Here, $F_{\text{Jupiter closest }}$ is the gravitational force of Jupiter on the earth when it is closest to earth,  $M_{\text{Jupiter }}$ is the mass of Jupiter, and $r_{\text{Jupiter }}$ is the radius of Jupiter.

Write the expression for gravitational force of Jupiter on the earth when it is furthest to earth.

    $F_{\text{Jupiter Furthest }}=\frac{G{M}_{\text{Jupiter }}{M}_{\text{Earth }}}{{\left(r_{\text{Jupiter }}+r_{\text{Earth }}\right)}^2}$        (III)

Here, $F_{\text{Jupiter Furthest }}$ is the gravitational force of Jupiter on the earth when it is furthest to earth, $M_{\text{Jupiter }}$ is the mass of Jupiter, and $r_{\text{Jupiter }}$ is the radius of Jupiter.

Conclusion:

Substitute $6.67\times {10}^{-11}\text{N}\cdot {\text{m}}^2/\text{kg}$ for $G$, $1.99\times {10}^{30}\text{kg}$ for $M_{\text{sun}}$, $1.50\times {10}^{11}\text{m}$ for $r_{\text{E}}$ and $5.97\times {10}^{24}\text{kg}$ for $M_{\text{Earth}}$ in equation (I), to find $F_{\text{SE}\text{grav}}$.

    $\begin{array}{c}{F}_{\text{SE}\text{grav}}=\frac{\left(6.67\times {10}^{-11}\text{N}\cdot {\text{m}}^2/\text{kg}\right)\left(1.99\times {10}^{30}\text{kg}\right)\left(5.97\times {10}^{24}\text{kg}\right)}{{\left(1.50\times {10}^{11}\text{m}\right)}^2}\\ =3.5\times {10}^{22}\text{N}\end{array}$

Substitute $6.67\times {10}^{-11}\text{N}\cdot {\text{m}}^2/\text{kg}$ for $G$, $1.99\times {10}^{27}\text{kg}$ for $M_{\text{Jupiter}}$, $1.50\times {10}^{11}\text{m}$ for $r_{\text{Earth}}$, $7.78\times {10}^{11}\text{m}$ for $r_{\text{Jupiter}}$ and $5.97\times {10}^{24}\text{kg}$ for $M_{\text{Earth}}$ in equation (II), to find $F_{\text{Jupiter closest }}$.

    $\begin{array}{c}{F}_{\text{Jupiter closest }}=\frac{\left(6.67\times {10}^{-11}\text{N}\cdot {\text{m}}^2/\text{kg}\right)\left(1.99\times {10}^{27}\text{kg}\right)\left(1.50\times {10}^{11}\text{m}\right)}{{\left(\left(7.78\times {10}^{11}\text{m}\right)-\left(5.97\times {10}^{24}\text{kg}\right)\right)}^2}\\ =1.9\times {10}^{18}\text{N}\end{array}$

Substitute $6.67\times {10}^{-11}\text{N}\cdot {\text{m}}^2/\text{kg}$ for $G$, $1.99\times {10}^{27}\text{kg}$ for $M_{\text{Jupiter}}$, $1.50\times {10}^{11}\text{m}$ for $r_{\text{Earth}}$, $7.78\times {10}^{11}\text{m}$ for $r_{\text{Jupiter}}$ and $5.97\times {10}^{24}\text{kg}$ for $M_{\text{Earth}}$ in equation (III), to find $F_{\text{Jupiter furthest}}$.

    $\begin{array}{c}{F}_{\text{Jupiter furthest}}=\frac{\left(6.67\times {10}^{-11}\text{N}\cdot {\text{m}}^2/\text{kg}\right)\left(1.99\times {10}^{27}\text{kg}\right)\left(1.50\times {10}^{11}\text{m}\right)}{{\left(\left(7.78\times {10}^{11}\text{m}\right)+\left(5.97\times {10}^{24}\text{kg}\right)\right)}^2}\\ =8.8\times {10}^{17}\text{N}\end{array}$

Therefore, the gravitational force of Jupiter on the earth when it is closest to earth is $\underset{\_}{1.9\times {10}^{18}\text{N}}$, The gravitational force of Jupiter on the earth when earth is farthest to earth is $\underset{\_}{8.8\times {10}^{17}\text{N}}$, and the gravitational force of the sun on the earth is $\underset{\_}{3.5\times {10}^{22}\text{N}}$.