Question

The formula to find the period of orbit of a satellite around a planet is T2=(4π2GM)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

Answer:

$r=\sqrt[3]{\dfrac{T^2GM}{4\pi^2}}$

Step-by-step explanation:

Divide by the coefficient of the r factor, then take the cube root.

$T^2=\dfrac{4\pi^2}{GM}r^3 \qquad\text{given formula}\\\\\dfrac{T^2GM}{4\pi^2}=r^3 \qquad\text{divide by the coefficient of the r factor}\\\\r=\sqrt[3]{\dfrac{T^2GM}{4\pi^2}} \qquad\text{cube root}$

Answer 2

Answer:

The formula to solve r is $r=\sqrt[3]{\frac{GMT^{2}}{4\pi^{2}}}$.

Step-by-step explanation:

Consider the provided formula:

$T^{2}=\frac{4\pi^{2}r^{3}}{GM}$

Where r is the orbit’s mean radius, M is the mass of the planet, and G is the universal gravitational constant.

Multiply both side by GM.

$T^{2}GM=4\pi^{2}r^{3}$

Further solve the above equation.

$\frac{T^{2}GM}{4\pi^{2}}=r^{3}$

$\sqrt[3]{\frac{GMT^{2}}{4\pi^{2}}}=r$

Hence, the formula to solve r is $r=\sqrt[3]{\frac{GMT^{2}}{4\pi^{2}}}$.