Answer:
[tex]G=\frac{4\pi^2r_m^3}{T^2M_e}[/tex]
Explanation:
We write the force of gravity a body of mass [tex]m[/tex] experiments by being at a distance [tex]r_m[/tex] from a larger body (the earth) of mass [tex]M_E[/tex] using Newton's law of universal gravitation:
[tex]F=\frac{GM_em}{r_m^2}[/tex]
Since this force is the centripetal force that keeps the object on orbit, and the velocity of the orbiting body will be given by the relation between the circumference [tex]2\pi r_m[/tex] and the period [tex]T[/tex] of the orbit, we can write:
[tex]F=ma_{cp}=m\frac{v^2}{r_m}=\frac{m}{r_m}(\frac{2\pi r_m}{T})^2=\frac{4\pi^2mr_m}{T^2}[/tex]
Which means:
[tex]\frac{GM_em}{r_m^2}=\frac{4\pi^2mr_m}{T^2}[/tex]
or:
[tex]G=\frac{4\pi^2r_m^3}{T^2M_e}[/tex]
