The gravitational force formula is [tex]F=\frac{G m m_2}{r^2}[/tex], where [tex]F[/tex] is the force between two objects, [tex]G[/tex] is the constant 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 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{ Gm _2 m_2}}{F}[/tex]
B. [tex]r=\frac{\sqrt{G m m_2 F}}{F}[/tex]
C. [tex]r=\sqrt{G m_1 m_2 F}[/tex]
Answer (2)
The correct expression for the distance r in simplest form is given by Option B: r = F G m 1 m 2 F . This matches with our derived formula after simplification. Hence, the answer is Option B.
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Start with the given formula: r = F G m 1 m 2 .
Multiply the numerator and denominator inside the square root by F : r = F 2 G m 1 m 2 F .
Simplify by taking the square root of the denominator: r = F G m 1 m 2 F .
The simplified expression matches option 2: r = F G m m 2 F .
Explanation
Understanding the Formula We are given the formula for the distance between two objects based on gravitational force: r = F G m 1 m 2 , where F is the gravitational force, G is the gravitational constant, m 1 and m 2 are the masses of the objects, and r is the distance between them. Our goal is to simplify this expression and match it to one of the given options.
Multiplying by F F To simplify the expression, we can multiply the numerator and the denominator inside the square root by F . This will help us get rid of the fraction inside the square root and potentially simplify the expression further:
r = F G m 1 m 2 ⋅ F F = F 2 G m 1 m 2 F
Taking the Square Root of the Denominator Now, we can take the square root of the denominator, since F 2 = F . This gives us:
r = F 2 G m 1 m 2 F = F G m 1 m 2 F
Matching the Expression Comparing this simplified expression with the given options, we see that it matches the second option:
r = F G m 1 m 2 F
Examples
Understanding gravitational force and distance is crucial in many real-world applications. For example, calculating the orbital distances of satellites around the Earth, determining the distance between stars in a galaxy, or even designing experiments to measure gravitational forces in a lab. Simplifying formulas like this allows scientists and engineers to make quick and accurate calculations, which are essential for space exploration, astrophysics, and other fields.
