Chapter 4, Problem 35Q

To determine

The gravitational force exerted by Earth on the Moon and by Moon on Earth, when the mass of Earth is given as $5.98\times {10}^{24}\text{ kg}$, mass of Moon is given as $7.35\times {10}^{22}\text{ kg}$ and the distance between them is 38400 km. Also, compare the force calculated with the gravitational force between the Sun and Earth using values from the text.

Answer to Problem 35Q

Solution:

$1.98\times {10}^{20}\text{ N}$, $1.98\times {10}^{20}\text{ N}$ and $\frac{F_{Sun-Earth}}{F_{Earth-Moon}}=178.28$

Explanation of Solution

Given data:

Mass of the Moon is $7.35\times {10}^{22}\text{ kg}$ and mass of Earth is $5.98\times {10}^{24}\text{ kg}$.

The average distance of center of Earth and center of Moon is 384400 km.

From the text, the gravitational force $F_{\text{Sun-Earth}}$ between the Sun and the Earth is $3.53\times {10}^{22}\text{ N}$.

Formula used:

Newton’s law of Universal gravitation can be stated by an equation as:

$F=G\left(\text{}\frac{m_1{m}_2}{r^2}\right)$

Here, $m_1$ is the mass of the first object, $m_2$ is the mass of the second object, r is the distance between the objects and G is the universal constant of gravitation having a value of $6.67\times {10}^{-11}\text{ N}\cdot {\text{m}}^{\text{2}}{\text{/kg}}^{\text{2}}$.

Explanation:

From Newton’s law of gravitation, the gravitational force, F is proportional to the mass and inversely proportional to the square of the distance r.

Use Newton’s equation and write the gravitational force of Earth and Moon:

$F_{Earth-Moon}=G\left(\frac{m_{Moon}{m}_{Earth}}{r^2}\right)$

Distance between Earth and Moon is 384400 km. Convert the unit of distance to m.

$\begin{array}{c}\text{1 km = 1000 m}\\ \text{384400 km = 384400}\left(1000\right)\text{ m}\\ =3.844\times {10}^8\text{ m}\end{array}$

Substitute $5.98\times {10}^{24}\text{ kg}$ for mass of Earth, $7.35\times {10}^{22}\text{ kg}$ for mass of Moon, $6.67\times {10}^{-11}\text{ N}\cdot {\text{m}}^{\text{2}}{\text{/kg}}^{\text{2}}$ for G and $3.844\times {10}^8\text{ m}$ for r.

$\begin{array}{c}{F}_{Earth-Moon}=\left(6.67\times {10}^{-11}\text{ N}\cdot {\text{m}}^{\text{2}}{\text{/kg}}^{\text{2}}\right)\left(\frac{\left(5.98\times {10}^{24}\text{ kg}\right)\left(7.35\times {10}^{24}\text{ kg}\right)}{{\left(3.844\times {10}^8\text{ m}\right)}^2}\right)\\ =1.98\times {10}^{20}\text{ N}\end{array}$ …… (1)

Hence, the gravitational force between Earth and Moon is $1.98\times {10}^{20}\text{ N}$.

Observe that the force exerted by Earth on Moon is equal to the force exerted by Moon on Earth, according to Newton’s law of gravity.

Observe from equation 1 and compare the force between Earth and Moon, $F_{Earth-Moon}$ and force between the Sun and Earth, $F_{\text{Sun-Earth}}$.

$\begin{array}{c}\frac{F_{Sun-Earth}}{F_{Earth-Moon}}=\frac{3.53\times {10}^{22}\text{ N}}{1.98\times {10}^{20}\text{ N}}\\ =178.28\end{array}$

Conclusion:

Therefore, the force of the Sun on Earth is approximately 178 times more than the force of Earth on the Moon. This is majorly due the huge mass of the Sun as compared to the Moon, and also because of the large distance between the Sun and Earth as compared to the Moon and Earth.