Solved Problem on Kepler's Laws and Gravitation

In the film 2001: A Space Odyssey (1968 directed by Stanley Kubrick) the spacecraft Discovery One has a section formed by a centrifuge that rotates in order to produce artificial gravity similar to the gravity of the Moon.

Assuming that an astronaut has an average height of 1.70 m and that the centrifuge has a diameter of 11.6 m and rotates at a frequency of 5 rpm, check if it is possible to build such a device.

Problem data:
  • Centrifuge diameter:    D = 11.6 m;
  • Centrifuge rotation frequency:    f = 5 rpm;
  • Astronaut height:    h = 1.70 m.
Problem diagram:

Since the centrifuge has a diameter of 11.6 m, its radius is
\[ \begin{gather} R=\frac{D}{2}\\ R=\frac{11.6}{2}\\ R=5.8\;\text{m} \end{gather} \]
this is the distance from the feet of the astronaut to the center of the centrifuge.
The head of the astronaut is at a distance d from the center equal to
\[ \begin{gather} d=R-h\\ d=5.8-1.7\\ d=4.1\;\text{m} \end{gather} \]
With these data, we can calculate the artificial gravity generated by rotating the centrifuge at the height of the feet of the astronaut, \( \vec{g}_{f} \), and head, \( \vec{g}_{h} \) (Figure 1).
Figure 1


First, we must convert the given frequency in revolutions per minute (rpm) to hertz (Hz) used in the International System of Units (SI)
\[ \begin{gather} f=5\;\text{rpm}=5\;\frac{\text{rotations}}{1\;\cancel{\text{min}}}\times\frac{1\;\cancel{\text{min}}}{60\;\text{s}}=\frac{5}{60}\;\frac{\text{rotations}}{\text{s}}=\frac{1}{12}\;\frac{\text{rotations}}{\text{s}}=\frac{1}{12}\;\text{Hz} \end{gather} \]

Forces acting on the astronaut (Figure 2):
  • \( {\vec{F}}_{CP} \): centripetal force acting on the astronaut due to the rotation of the centrifuge;
  • \( {\vec{F}}_{CG} \): centrifugal force that acts on the astronaut, keeps the astronaut on the floor of the centrifuge producing artificial gravity.
Note: The centrifugal force, \( {\vec{F}}_{CG} \), is not the reaction of the centripetal force, \( {\vec{F}}_{CP} \), by Newton's Third Law, action and reaction forces are on different bodies, and in this case, the two forces are applied to the astronaut.

Figure 2

The artificial gravity produced by the rotation is given by the centrifugal force and is equal to the centripetal force, in magnitude, given by
\[ \begin{gather} F_{CG}=F_{CP}=m\frac{v^{2}}{r} \end{gather} \]
Applying Newton's Second Law
\[ \begin{gather} \bbox[#99CCFF,10px] {\vec{F}=m\vec{a}} \end{gather} \]
the resultant force that produces artificial gravity is the centrifugal force, so we write its magnitude
\[ \begin{gather} m\frac{v^{2}}{r}=ma\\ a=\frac{v^{2}}{r} \tag{I} \end{gather} \]
the tangential speed is given by
\[ \begin{gather} \bbox[#99CCFF,10px] {v=\omega r} \tag{II} \end{gather} \]
substituting expression (II) into expression (I)
\[ \begin{gather} a=\frac{(\omega r)^{2}}{r}\\ a=\frac{\omega^{2}r^{2}}{r}\\ a=\omega ^{2}r \tag{III} \end{gather} \]
The angular speed is given by
\[ \begin{gather} \bbox[#99CCFF,10px] {\omega =2\pi f} \tag{IV} \end{gather} \]
substituting expression (IV) into expression (III)
\[ \begin{gather} a=(2\pi f)^{2}r\\ a=4\pi^{2}f^{2}r \tag{V} \end{gather} \]
Using the given frequency, taking π = 3.14 and using r = R = 5.8 m, we have the acceleration produced at the height of the feet of the astronaut, a = gf
\[ \begin{gather} g_{f}=4\times 3.14^{2}\times \left(\frac{1}{12}\right)^{2}\times 5.8\\ g_{f}=4\times 9.86\times \frac{1}{144}\times 5.8\\ g_{f}=1.59\;\text{m/s}^{2} \end{gather} \]
Using r = d = 4.1 m, we have the acceleration generated at the height of the head of the astronaut, a = gh
\[ \begin{gather} g_{h}=4\times 3.14^{2}\left(\frac{1}{12}\right)^{2}\times 4.1\\ g_{h}=4\times 9.86.\frac{1}{144}\times 4.1\\ g_{h}=1.12\;\text{m/s}^{2} \end{gather} \]
The difference in values ​​between the acceleration due to gravity in the head and feet is very large, this would make it difficult for astronauts to walk, blood circulation could also be compromised, with a lack of blood in the head, which would be attracted to the feet, causing vertigo. and fainting of the astronauts. Therefore, it would not be possible to build a centrifuge with these dimensions.