The gravitational force formula is [tex]F=\frac{G m_1 m_2}{r^2}[/tex], where [tex]F[/tex] is the force between two objects, [tex]G[/tex] is the constant of gravitation, [tex]m_1[/tex] is the mass of the first object, [tex]m_2[/tex] is the mass of the second object, and [tex]r[/tex] is the distance between the objects. By rewriting the formula as [tex]r=\sqrt{\frac{G m_1 m_2}{F}}[/tex], you can find the distance between objects. Which of the following gives the distance, [tex]r[/tex], in simplest form?
A. [tex]r=\frac{\sqrt{G m_1 m_2}}{F}[/tex]
B. [tex]r=\frac{\sqrt{G m_1 m_2 F}}{F}[/tex]
C. [tex]r=\sqrt{G m_1 m_2 F}[/tex]
Answer (2)
The distance r between two objects can be derived from the gravitational force formula. By simplifying r = F G m 1 m 2 , we find that the correct choice that represents this distance is option B: r = F G m 1 m 2 F .
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Rewrite the expression as a fraction of square roots: r = F G m 1 m 2 .
Multiply the numerator and denominator by F to rationalize the denominator.
Combine the square roots in the numerator: r = F G m 1 m 2 F .
The simplified expression is r = F G m 1 m 2 F .
Explanation
Understanding the Problem We are given the formula for the distance between two objects based on gravitational force: r = F G m 1 m 2 . Our goal is to simplify this expression and match it to one of the provided options.
Separating the Square Root To simplify the expression, we can rewrite the square root of a fraction as a fraction of square roots: r = F G m 1 m 2
Rationalizing the Denominator To rationalize the denominator, we multiply both the numerator and the denominator by F : r = F G m 1 m 2 ⋅ F F = F G m 1 m 2 ⋅ F
Combining Square Roots Now, we can combine the square roots in the numerator: r = F G m 1 m 2 F
Final Answer Comparing this simplified expression with the given options, we find that it matches the second option: r = F G m 1 m 2 F .
Examples
Understanding gravitational force and how it relates to distance is crucial in many real-world applications. For example, when planning satellite orbits, engineers must calculate the precise distance a satellite needs to be from Earth to maintain a stable orbit. This calculation involves using the gravitational force formula and solving for the distance, r . Similarly, astronomers use this formula to determine the distances between stars and planets in our galaxy, helping them understand the structure and dynamics of celestial bodies. By simplifying the formula, we make these calculations more manageable and accurate.
