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 start with the formula \( r = \sqrt{\frac{Gm_1m_2}{F}} \). To simplify the square root of a fraction, we can rationalize the denominator by multiplying the numerator and the denominator inside the square root by \( F \) to make the denominator a perfect square (since \( F\times F = F^2 \), and \( \sqrt{F^2}=F \)). So we have:
\[
r=\sqrt{\frac{Gm_1m_2\times F}{F\times F}}
\]

Step2: Simplify the Square Root

Now, we can simplify the square root. The denominator inside the square root is \( F^2 \), so \( \sqrt{\frac{Gm_1m_2F}{F^2}}=\frac{\sqrt{Gm_1m_2F}}{\sqrt{F^2}} \). Since \( \sqrt{F^2} = F \) (assuming \( F>0 \), which makes sense in the context of force), we get:
\[
r = \frac{\sqrt{Gm_1m_2F}}{F}
\]

Answer:

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