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

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

Explanation:

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

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

Answer 2

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

Explanation:

Consider the provided formula:

`T^(2)=(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.

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

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

Hence, the formula to solve r is
`r=\sqrt[3]{(GMT^(2))/(4\pi^(2))}`.