Chapter 10, Problem 21Q

(a)

To determine

The distance between the center of mass of the Earth-Moon system and the center of Earth using the expression, $d_{cm}=\frac{m_{2}r}{m_1+m_2}$, considering Earth as world 1 and the Moon as world 2, where $r$ represents the center-to-center distance between the two worlds.

Answer to Problem 21Q

Solution:

$4.67\times {10}^6\text{ m}$.

Explanation of Solution

Introduction:

The distance between the center of mass of a two worlds system and the center of world 1 $\left(d_{cm}\right)$ is related to their respective masses $\left(m_1\text{ and }{m}_2\right)$ and the center-to-center distance between the two worlds $\left(r\right)$ as:

$d_{cm}=\frac{m_{2}r}{m_1+m_2}$

Explanation:

Recall the expression for the distance between the center of mass of the Earth-Moon system and the center of Earth, considering Earth as world 1 and the Moon as world 2, and their respective centers being an average distance apart from each other as:

$d_{cm}=\frac{m_{2}r}{m_1+m_2}$

Substitute $7.35\times {10}^{22}\text{ kg}$ for $m_2$, $3.84\times {10}^8\text{ m}$ for $r$, and $5.97\times {10}^{24}\text{ kg}$ for $m_1$.

$\begin{array}{l}{d}_{cm}=\frac{\left(7.35\times {10}^{22}\text{ kg}\right)\left(\text{3}\text{.84}\times {\text{10}}^8\text{ m}\right)}{\left(5.97\times {10}^{24}\text{ kg}+7.35\times {10}^{22}\text{ kg}\right)}\\ d_{cm}=4.67\times {10}^6\text{ m}\end{array}$

Conclusion:

Hence, the distance between the center of mass of the Earth-Moon system and the center of Earth is $4.67\times {10}^6\text{ m}$.

(b)

To determine

Whether the center of mass of the Earth-Moon system is beneath the earth’s surface and if yes, then also determine the depth of it from the surface of Earth.

Answer to Problem 21Q

Solution:

The center of mass of the Earth-Moon system is within the Earth, at a distance of about $1710\text{ km}$ from the surface of Earth.

Explanation of Solution

Introduction:

The center of mass of the Earth-Moon system lies on the imaginary line joining the centers of both the worlds, and since the mass of Earth is much more than that of the Moon, the center of mass will lie closer to Earth’s center and beneath its surface.

Therefore, the distance between the center of mass of the system and the Earth’s surface is:

$d=R_E-r$

Here, $R_E$ represents the radius of Earth, and $r$ represents the distance between the center of mass of the Earth-Moon system and Earth’s center.

Explanation:

Refer to the sub-part (a) of the problem and compare the value of the distance of the center of mass of the Earth-Moon system with the standard value of Earth’s radius. It is observed that the radius of Earth is much larger than the distance of the center of mass of the system. This can be represented as, $r<<R_E$.

Hence, the center of mass of the system will lie within the Earth, below its surface.

Now, recall the expression for the distance between the center of mass of the system and Earth’s surface as:

$d=R_E-r$

Substitute $6.38\times {10}^6\text{ m}$ for $R_E$ and $4.67\times {10}^6\text{ m}$ for $r$.

$\begin{array}{l}d=6.38\times {10}^6\text{ m}-4.67\times {10}^6\text{ m}\\ d=1.71\text{ m}\left(\frac{1\text{ km}}{1000\text{ m}}\right)\\ d=1710\text{ km}\end{array}$

Conclusion:

Hence, the center of mass of the Earth-Moon system lies beneath the surface of Earth at a distance of about $1710\text{ km}$.

(c)

To determine

The distance between the center of the Sun and the center of mass of the Sun-Earth system, if the first world is the Sun and the second world is Earth using the expression, $d_{cm}=\frac{m_{2}r}{m_1+m_2}$. Also compare the result with the radius of the Sun, and use this result to discern whether it is safe to assume that Earth orbits around the center of the Sun.

Answer to Problem 21Q

Solution:

Distance of the center of mass of the Sun-Earth system is around $4.49\times {10}^5\text{ m}$, which is about $1\%$ of the radius of the Sun ($6.96\times {10}^8\text{ m}$). Thus, it is safe to assume that Earth rotates around the center of the Sun.

Explanation of Solution

Introduction:

The distance between the center of mass of a two worlds system and the center of world 1 $\left(d_{cm}\right)$ is related to their respective masses $\left(m_1\text{ and }{m}_2\right)$ and the center-to-center distance of the two worlds $\left(r\right)$ as:

$d_{cm}=\frac{m_{2}r}{m_1+m_2}$

Explanation:

Recall the expression for the distance between the center of mass of the Sun-Earth system and the center of the Sun, considering the Sun as world 1 and Earth as world 2:

$d_{cm}=\frac{m_{2}r}{m_1+m_2}$

Substitute $5.972\times {10}^{24}\text{ kg}$ for $m_2$, $1.989\times {10}^{30}\text{ kg}$ for $m_1$, and $1.496\times {10}^{11}\text{ m}$ for $r$.

$\begin{array}{c}{d}_{cm}=\frac{\left(5.972\times {10}^{24}\text{ kg}\right)\left(\text{149}\text{.6}\times {\text{10}}^{11}\text{ m}\right)}{\left(5.972\times {10}^{24}\text{ kg}\right)+\left(1.989\times {10}^{30}\text{ kg}\right)}\\ =4.49\times {10}^5\text{ m}\end{array}$

Compare the value calculated above with the standard value of the radius of the Sun, which is $6.96\times {10}^8\text{ m}$.

It is observed that the value of the distance between the center of mass of the system and the center of the Sun is almost $1\%$ of the radius of the Sun and therefore can be assumed to be at the center of the Sun.

Therefore, it can be safely assumed that Earth revolves around the center of the Sun.

Conclusion:

Hence, the distance between the center of mass of the Sun-Earth system and the center of the Sun is around $4.49\times {10}^5\text{ m}$, which is negligible in comparison to the radius of the Sun.

Therefore, it is safe to assume that Earth revolves around the Sun’s center.