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
