Collection of Solved Problems in Physics

The Earth-Moon System

Task number: 2233

• Notation

  m	mass of the Moon
  M = 81m	mass of the Earth
  a = 384 000 km	distance between the centre of the Moon and centre of the Earth
  xm = ?	distance between the centre of mass of the system and the centre of the Earth

• Hint 1

  Draw the above-mentioned situation in an appropriate coordinate system. Which coordinates of the centre of mass of the system require calculation and which ones can be determined right from the figure without any calculation?

• Solution of Hint 1

  The given system is shown in the figure below. The coordinate system is chosen in a way that its origin lies in the centre of the Earth and its x-axis is passing through the centre of the Moon.

  

  Since the centres of mass of both considered bodies lie on the x-axis, centre of mass of their system will lie on the x-axis as well. All points of the x-axis have zero y- and z-coordinates, which means that for the coordinates y_m and z_m of the centre of mass it holds true:

  \[y_m\,=\,z_m\,=\,0.\]

  Therefore, we just need to calculate the remaining coordinate of the centre of mass x_m.

• Hint 2

  In the given situation, we can consider Earth and Moon to be point masses because the distance between them (384 000 km) is many times greater than their dimensions (radius of the Earth is 6378 km). Therefore, in order to calculate the unknown coordinate x_m, we can use the formula for the centre of mass of the system of point masses. How does this formula look like?

• Solution of Hint 2

  Formula for the position of the centre of mass of a system with n mass points is as follows:

  \[\vec{r}_m\,=\,\frac{\sum_{i=1}^n{m_i\vec{r}_i}}{\sum_{i=1}^n{m_i}},\]

  where m_i is the mass of each mass point and r_i is the associated position vector.Since the coordinates of the centre of mass can be calculated separately, for coordinate x_m in our system we get:

  \[x_m\,=\,\frac{\sum_{i=1}^2{m_ix_i}}{\sum_{i=1}^2{m_i}}.\tag{1}\]

• Hint 3

  Substitute the given values into relation (1) and calculate it. Express the distance between the centre of mass of the system and the centre of the Earth as a proportion of the Earth's radius.

• Solution of Hint 3

  After substitution into relation (1), we obtain:

  \[x_m\,=\,\frac{M{\cdot}0\,+\,m{\cdot}a}{M\,+\,m}\,=\,\frac{ma}{81m\,+\,m}\,=\,\frac{a}{82}.\]

  Číselně:

  \[x_m\,=\,\frac{384\,000}{82}\,\mathrm{km}\,\dot=\,4683\,\mathrm{km}.\]

  Since the radius of the Earth is 6378 km, distance x_m makes approximately 73.4% of this value. Thus, the centre of mass of the Earth-Moon system is located below the Earth's surface (we can see the figure is not consistent with this fact).

• Overall Solution

  First, we draw a picture of the situation (see below). The Earth and the Moon can be considered mass points because the distance between them (384 000 km) is many times greater than their dimensions (radius of the Earth is 6378 km). The coordinate system is chosen in a way that its origin lies in the centre of the Earth and its x-axis is passing through the centre of the Moon:

  

  Since the centres of mass of both considered bodies lie on the x-axis, centre of mass of their system will lie on the x-axis as well. All points of the x-axis have zero y-ovou and z-coordinates, which means that for the coordinates y_m and z_m of the centre of mass the following holds true:

  \[y_m\,=\,z_m\,=\,0.\]

  Therefore, in order to calculate the unknown coordinate x_m, we can use the formula for the centre of mass of a system with n mass points:

  \[\vec{r}_m\,=\,\frac{\sum_{i=1}^n{m_i\vec{r}_i}}{\sum_{i=1}^n{m_i}},\]

  where m_i is the mass of each mass point and r_i is the associated position vector. Since the coordinates of the centre of mass can be calculated separately, for coordinate x_m in our system we get:

  \[x_m\,=\,\frac{\sum_{i=1}^2{m_ix_i}}{\sum_{i=1}^2{m_i}}.\tag{1}\]

  After substitution into relation (1), we obtain:

  \[x_m\,=\,\frac{M{\cdot}0\,+\,m{\cdot}a}{M\,+\,m}\,=\,\frac{ma}{81m\,+\,m}\,=\,\frac{a}{82}.\]

  Numerically:

  \[x_m\,=\,\frac{384\,000}{82}\,\mathrm{km}\,\dot=\,4683\,\mathrm{km}.\]

  Since the radius of the Earth is 6378 km, distance x_m makes approximately 73.4% of this value. Thus, the centre of mass of the Earth-Moon system is located below the Earth's surface (we can see the figure is not consistent with this fact).

• Answer

  Position of the centre of mass of the Earth-Moon system is determined in our coordinate system by the following coordinates:

  \[x_m\,\dot=\,4683\,\mathrm{km},\] \[y_m\,=\,0\,\mathrm{km},\] \[z_m\,=\,0\,\mathrm{km}.\]

  Centre of mass of the system is located at the distance of 0.734 multiple of the Earth's radius from the centre of the Earth.