The formula to find the period of orbit of a satellite around a planet is T^2=(4pi^2/GM)r^3 where r is the orbit's mean radius, M is the mass of the planet, and G is the universal gravitational constant. if you are given all the values except r, how do you rewrite the formula to solve for r?

Respuesta :

The answer is [tex]r= \sqrt[3]{GMT^{2}/4 \pi^{2}} [/tex]

[tex] T^{2} = \frac{4 \pi^{2}}{GM} r^{3} [/tex]

Move [tex]\frac{4 \pi^{2} }{GM} [/tex] to the other side of the equation:
[tex] T^{2} /\frac{4 \pi^{2} }{GM} = r^{3} \\ T^{2} *\frac{GM}{4 \pi^{2} } = r^{3} [/tex]

Rearrange:
[tex]r^{3} = T^{2} *\frac{GM}{4 \pi^{2} } \\ r^{3}= \frac{T^{2} *GM}{4 \pi^{2} } \\ r^{3}= \frac{GMT^{2}}{4 \pi^{2} } \\ r^{3} = GMT^{2}/4 \pi^{2}[/tex]

Since [tex] x^{3}= \sqrt[3]{x} [/tex], then
[tex]r^{3} = GMT^{2}/4 \pi^{2} \\ r= \sqrt[3]{GMT^{2}/4 \pi^{2}} [/tex]