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Q109P

Expert-verifiedFound in: Page 255

Book edition
10th Edition

Author(s)
David Halliday

Pages
1328 pages

ISBN
9781118230718

**(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?**

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

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

**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.**

${X}_{com}=\frac{({m}_{1}{x}_{1}+{m}_{2}{x}_{2})}{{m}_{1}+{m}_{2}}$

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.

${X}_{com}=\frac{(5.98\times {10}^{24})\left(0\right)+(7.36\times {10}^{22})(3.82\times {10}^{8})}{(5.98\times {10}^{24})+(7.36\times {10}^{22})}\phantom{\rule{0ex}{0ex}}=4.64\times {10}^{6}\mathrm{m}\phantom{\rule{0ex}{0ex}}=4.6\times {10}^{3}\mathrm{km}$

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

We get,

$\frac{(4.64\times {10}^{6})}{6.37\times {10}^{6}}=0.73\phantom{\rule{0ex}{0ex}}=73\%$

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

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