Multiple Choice
In Newton's law of universal gravitation, , which two factors determine the magnitude of the gravitational force between two objects?
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8. Centripetal Forces & Gravitation
Newton's Law of Gravity
Multiple Choice
According to Newton's law of universal gravitation, if the distance between two point masses increases from to while the masses remain the same, how does the gravitational force change?
A
It becomes one-half as large: .
B
It becomes twice as large: .
C
It remains the same: .
D
It becomes one-fourth as large: .
Verified step by step guidance1
Recall Newton's law of universal gravitation, which states that the gravitational force \(F\) between two point masses is given by the formula:
\[F = G \frac{m_1 m_2}{r^2}\]
where \(G\) is the gravitational constant, \(m_1\) and \(m_2\) are the masses, and \(r\) is the distance between the masses.
Identify that the masses \(m_1\) and \(m_2\) remain constant, so the only variable changing is the distance \(r\), which increases to \$2r$.
Substitute the new distance \$2r\( into the formula to find the new force \)F'$:
\[F' = G \frac{m_1 m_2}{(2r)^2}\]
Simplify the denominator:
\[(2r)^2 = 4r^2\]
so the new force becomes:
\[F' = G \frac{m_1 m_2}{4r^2}\]
Compare the new force \(F'\) to the original force \(F\) by dividing \(F'\) by \(F\):
\[\frac{F'}{F} = \frac{G \frac{m_1 m_2}{4r^2}}{G \frac{m_1 m_2}{r^2}} = \frac{1}{4}\]
This shows that the gravitational force becomes one-fourth as large when the distance doubles.
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