Multiple Choice
In an experiment, which of the following measurements depends directly on the gravitational force (and therefore changes when moving from Earth to the Moon)?
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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 the masses remain the same, how does the gravitational force change?
A
It remains the same.
B
It becomes one-fourth as large.
C
It becomes one-half as large.
D
It becomes twice 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 \(m_1\) 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 the same, but the distance \(r\) between them is doubled, so the new distance is \$2r$.
Substitute the new distance into the formula to find the new force \(F_{new}\):
\[F_{new} = G \frac{m_1 m_2}{(2r)^2}\]
Simplify the denominator:
\[(2r)^2 = 4r^2\]
so the new force becomes
\[F_{new} = G \frac{m_1 m_2}{4r^2}\]
Compare the new force \(F_{new}\) to the original force \(F\):
\[F_{new} = \frac{1}{4} F\]
This shows that doubling the distance reduces the gravitational force to one-fourth of its original value.
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