Question

(a) How far is the center of mass of the Earth–Moon system from the center of Earth? (Appendix C gives the masses of Earth and the Moon and the distance between the two.) (b) What percentage of Earth’s radius is that distance?

Short answer

1. Position of center of mass of the Earth-Moon system,$X_{com}\mathrm{is}4.6\times {10}^3\mathrm{km}$.
2. Percentage of that distance with Earth’s radius is 73% .

Step-by-step solution

Understanding the given information

Appendix C in the book gives the masses of Earth, Moon and distance between them.

Concept and formula used in the given question

Use the concept of center of mass. Using the equation of center of mass of position you find the position of center of mass of Earth –Moon system from the center of earth. Then find that distance in terms of Earth’s radius. The formula used is given below.

$X_{com}=\frac{(m_1{x}_1+m_2{x}_2)}{m_1+m_2}$

(a) Calculate the how far is the center of mass of the Earth–Moon system from the center of Earth

Consider origin at the center of earth and write as:

$X_{com}=\frac{M{x}_1+M{x}_2}{M+m}$

Where M is the mass of Earth and m is the mass of the moon.

$$\begin{gathered} X_{com}=\frac{(5.98\times {10}^{24})\left(0\right)+(7.36\times {10}^{22})(3.82\times {10}^8)}{(5.98\times {10}^{24})+(7.36\times {10}^{22})} \\ =4.64\times {10}^6\mathrm{m} \\ =4.6\times {10}^3\mathrm{km} \end{gathered}$$

(b) Calculate the percentage of Earth’s radius to that distance

Consider radius of earth is$r_E=6.37\times {10}^6\mathrm{m}$

We get,

$$\begin{gathered} \frac{(4.64\times {10}^6)}{6.37\times {10}^6}=0.73 \\ =73\% \end{gathered}$$

So, the percentage of distance is 73% of radius of earth.