Question

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?

Answer 1

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


`T^(2) = (4 \pi^(2))/(GM) r^(3)`

Move
`(4 \pi^(2) )/(GM)` to the other side of the equation:

`T^(2) /(4 \pi^(2) )/(GM) = r^(3) \\ T^(2) *(GM)/(4 \pi^(2) ) = r^(3)`

Rearrange:

`r^(3) = T^(2) *(GM)/(4 \pi^(2) ) \\ r^(3)= (T^(2) *GM)/(4 \pi^(2) ) \\ r^(3)= (GMT^(2))/(4 \pi^(2) ) \\ r^(3) = GMT^(2)/4 \pi^(2)`

Since
`x^(3)= \sqrt[3]{x}`, then

`r^(3) = GMT^(2)/4 \pi^(2) \\ r= \sqrt[3]{GMT^(2)/4 \pi^(2)}`