Question

the gravitational force formula is $f = \frac{gm_1m_2}{r^2}$, where $f$ is the force between two objects, $g$ is the constant of gravitation, $m_1$ is the mass of the first object, $m_2$ is the mass of the second object, and $r$ is the distance between the objects. by rewriting the formula as $r = \sqrt{\frac{gm_1m_2}{f}}$, you can find the distance between objects. which of the following gives the distance, $r$, in simplest form?\
\bigcirc $r = \frac{\sqrt{gm_1m_2}}{f}$\
\bigcirc $r = \frac{\sqrt{gm_1m_2f}}{f}$\
\bigcirc $r = \sqrt{gm_1m_2f}$

Explanation:

Step1: Rationalize the denominator

We have \(r = \sqrt{\frac{Gm_1m_2}{F}}\). Using the property \(\sqrt{\frac{a}{b}}=\frac{\sqrt{a}}{\sqrt{b}}\) (\(a = Gm_1m_2\), \(b = F\)), we get \(r=\frac{\sqrt{Gm_1m_2}}{\sqrt{F}}\). Then, multiply the numerator and denominator by \(\sqrt{F}\) to rationalize the denominator: \(r=\frac{\sqrt{Gm_1m_2}\times\sqrt{F}}{\sqrt{F}\times\sqrt{F}}\).

Step2: Simplify the expression

Using the property \(\sqrt{a}\times\sqrt{b}=\sqrt{ab}\) (\(a = Gm_1m_2\), \(b = F\)) and \(\sqrt{F}\times\sqrt{F}=F\), we have \(r = \frac{\sqrt{Gm_1m_2F}}{F}\).

Answer:

\(r=\frac{\sqrt{Gm_1m_2F}}{F}\) (the second option)