Chapter 5, Problem 30P

At what distance from the Earth will a spacecraft traveling directly from the Earth to the Moon experience zero net force because the Earth and Moon pull in opposite directions with equal force?

To determine

The distance from the Earth where a spacecraft traveling directly from Earth to the Moon experience zero net force.

Answer to Problem 30P

Solution:

  $3.46\times {10}^8\text{m}$

Explanation of Solution

Given:

The Earth and Moon pull in opposite directions with equal forces.

Formula Used: F (gravitational force) $=\frac{G{m}_1{m}_2}{r^2}$

Where, $m_1\text{ and }{m}_2$ aremasses of objects.

r : the distance between $m_1\text{ and }{m}_2$ .

G : Universal gravitational constant.

Calculations: Let the zero point be $d_e$ the distance from Earth and $d_m$ distance from the Moon,

Using a 1 kg test mass at this point,

Considering the gravitational forces, $F_g$ on this mass from the Earth and from the Moon is equal.

  $F_g=\frac{G{M}_e\times 1}{d_e{}^2}=\frac{G{M}_m\times 1}{d_m{}^2}$

  $\frac{M_e}{d_e^2}=\frac{M_m}{d^2{}_m}$

Mass of Earth: $5.97\times {10}^{24}\text{kg}$

Mass of Moon: $7.35\times {10}^{22}\text{kg}$

  $\begin{array}{l}\frac{d_e}{d_m}=\sqrt{\frac{M_e}{M_m}}\\ \frac{d_e}{d_m}=\sqrt{\frac{5.97\times {10}^{24}}{7.35\times {10}^{22}}}\\ \frac{d_e}{d_m}=9\end{array}$

  $d_e=9d_m$

This means, the distance of the point is $\frac{9}{10}$ the distance of the moon from the Earth.

So,

  $\begin{array}{l}=0.9\times 3.84\times {10}^8\\ =3.46\times {10}^8\text{m}\end{array}$

Conclusion:

The distance from the Earth where a spacecraft traveling directly from Earth to the Moon experience zero net force is $3.46\times {10}^8\text{m}$ .