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There is a gravitational force between a sponge and a towel that are [tex]0.5 \, \text{m}[/tex] apart. If they were [tex]1.0 \, \text{m}[/tex] apart instead, how would the gravitational force change?

[tex]F = -G \frac{m_1 m_2}{d^2}[/tex]

A. It would be the same
B. Four times as much
C. A fourth as much
D. Half as much



Answer :

GinnyAnswer
Let's examine how the gravitational force changes with distance using the gravitational force formula:
[tex]\[ F = -G \frac{m_1 m_2}{d^2} \][/tex]

Initially, the distance between the sponge and the towel is \( d_1 = 0.5 \) meters, so the gravitational force is:
[tex]\[ F_1 = -G \frac{m_1 m_2}{(0.5)^2} \][/tex]

Now, if the distance between the sponge and the towel increases to \( d_2 = 1.0 \) meters, the new gravitational force becomes:
[tex]\[ F_2 = -G \frac{m_1 m_2}{(1.0)^2} \][/tex]

We need to understand how \( F_2 \) compares to \( F_1 \). To do this, we can express \( F_2 \) in terms of \( F_1 \):

First, note that:
[tex]\[ (1.0)^2 = 1 \][/tex]
and
[tex]\[ (0.5)^2 = 0.25 \][/tex]

So, we have:
[tex]\[ F_2 = -G \frac{m_1 m_2}{1} \][/tex]
and
[tex]\[ F_1 = -G \frac{m_1 m_2}{0.25} = 4 \cdot -G \frac{m_1 m_2}{1} = 4 \cdot F_2 \][/tex]

Thus, we can see:
[tex]\[ F_2 = \frac{F_1}{4} \][/tex]

This means that when the distance between the sponge and the towel is doubled from 0.5 meters to 1.0 meters, the gravitational force becomes one-fourth as much.

So, the correct answer is:
a fourth as much

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