Question

A satellite moves in a circle around the Earth. The radius of this circle is equal to one half of the radius of the moon's orbit. The satellite completes one revolution in how many lunar months ?

Answer 1

Kepler's third law:

$$\begin{gathered} T^2\alpha r^3 \\ {\left(\frac{T_2}{T_1}\right)}^2={\left(\frac{r_2}{r_1}\right)}^3 \\ \left(\frac{T_2}{T_1}\right)={\left(\frac{r_2}{r_1}\right)}^{\frac{3}{2}} \\ \frac{T_2}{1}={\left(\frac{{\displaystyle \frac{r_1}{2}}}{r_1}\right)}^{\frac{3}{2}} \\ {T}_2={\left(\frac{1}{2}\right)}^{\frac{3}{2}} \\ ={\left(2^{-1}\right)}^{\frac{3}{2}} \\ ={\left(2\right)}^{-\frac{3}{2}}\mathrm{lunar}\mathrm{months} \end{gathered}$$