Answer:
Assuming the given radius is 10⁶ km, the radius of the second planet is 1.31 * 10⁶ km.
Explanation:
Newton's law of gravity:
(1) [tex]F=\frac{GMm}{r^2}[/tex]
Centripetal force:
(2) [tex]F=m\omega^2 r[/tex]
On a circular orbit both forces must be equal:
(3) [tex]\frac{GMm}{r^2}=m\omega^2 r[/tex]
Solving for ω:
(4) [tex]\omega=\sqrt{\frac{GM}{r^3}}[/tex]
The period T is given by:
(5) [tex]T=\frac{2\pi}{\omega}=2\pi\sqrt{\frac{r^3}{GM}}[/tex]
Taking the ratio of two periods:
(6) [tex]\frac{T_1}{T_2}=\sqrt{\frac{r_1^3}{r_2^3}}[/tex]
Solving for r₁:
(7) [tex]r_1=(\frac{T_1}{T_2})^{\frac{2}{3}}r_2[/tex]
