Solved Problem on Kepler's Laws and Gravitation
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Determine the angular speed of a satellite around the Earth, assuming a circular orbit, as a function of the distance from the center of the Earth.


Problem diagram:

Let us assume that the following quantities are known, distance from Earth to satellite, RE, the mass of Earth, ME, and Newton's Universal Gravitational Constant, G.
Figure 1

Solution

The centripetal force for a satellite rotating around the Earth (Figure 1), is given by
\[ \begin{gather} \bbox[#99CCFF,10px] {{\vec{F}}_{cp}=m{\vec{a}}_{cp}} \tag{I} \end{gather} \]
where m is the mass of the satellite, the centripetal acceleration is given by
\[ \begin{gather} \bbox[#99CCFF,10px] {a_{cp}=\frac{v^{2}}{r}} \tag{II} \end{gather} \]
the tangential speed is given by
\[ \begin{gather} \bbox[#99CCFF,10px] {v=\omega r} \tag{III} \end{gather} \]
substituting expression (III) into expression (II)
\[ \begin{gather} a_{cp}=\frac{(\omega r)^{2}}{r}\\ a_{cp}=\frac{\omega^{2}r^{\cancel{2}}}{\cancel{r}}\\ a_{cp}=\omega^{2}r \tag{IV} \end{gather} \]
substituting expression (IV) into expression (I)
\[ \begin{gather} F_{cp}=m\omega ^{2}r \tag{V} \end{gather} \]
The only force acting on the satellite is the gravitational force of attraction between the Earth and the satellite given by Newton's Law of Universal Gravitation A única força atuando no satélite é força de atração gravitacional da Terra dada pela Lei da Gravitação Universal de Newton
\[ \begin{gather} \bbox[#99CCFF,10px] {F_{G}=G\frac{Mm}{r^{2}}} \tag{VI} \end{gather} \]
this force is the centripetal resultant, substituting expression (V) into expression (VI), where M = ME is the mass of the Earth, r = RE is the distance from the satellite to the center of the Earth
\[ \begin{gather} G\frac{M_{E}\cancel{m}}{R_{E}^{2}}=\cancel{m}\omega^{2}R_{E} \end{gather} \]
canceling the satellite mass m from both sides of the equation
\[ \begin{gather} G\frac{M_{E}}{R_{E}^{2}}=\omega ^{2}R_{E}\\ \omega^{2}=G\frac{M_{E}}{R_{E}^{3}} \end{gather} \]
\[ \begin{gather} \bbox[#FFCCCC,10px] {\omega =\sqrt{G\frac{M_{E}}{R_{E}^{3}}\;}} \end{gather} \]
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