Multiple Choice
Which of the following processes relies most directly on Newton's law of universal gravitation as the dominant force governing the motion?
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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 is doubled while their masses remain constant, how does the magnitude of the gravitational force change?
A
It becomes twice as large (multiplied by ).
B
It becomes four times as large (multiplied by ).
C
It becomes one-half as large (multiplied by ).
D
It becomes one-fourth as large (multiplied by ).
Verified step by step guidance1
Recall Newton's law of universal gravitation, which states that the gravitational force \(F\) between two point masses \(m_1\) and \(m_2\) separated by a distance \(r\) is given by the formula:
\[F = G \frac{m_1 m_2}{r^2}\]
where \(G\) is the gravitational constant.
Identify the variables that change in the problem: the masses \(m_1\) and \(m_2\) remain constant, but the distance \(r\) between the masses is doubled, so the new distance is \$2r$.
Substitute the new distance into the formula to find the new force \(F_{new}\):
\[F_{new} = G \frac{m_1 m_2}{(2r)^2} = G \frac{m_1 m_2}{4r^2}\]
Compare the new force \(F_{new}\) to the original force \(F\):
\[\frac{F_{new}}{F} = \frac{G \frac{m_1 m_2}{4r^2}}{G \frac{m_1 m_2}{r^2}} = \frac{1}{4}\]
Conclude that when the distance between two masses is doubled, the gravitational force becomes one-fourth as large, meaning it is multiplied by \(\frac{1}{4}\).
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