The gravitational force between two objects is determined by Newton's law of universal gravitation, which states that the gravitational force \( F \) 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 of the objects, and \( r \) is the distance between the centers of the two objects.
Initially, the two objects are 15 m apart. If one of them moves 3 m closer, the new distance between them becomes:
\[ r' = 15 m - 3 m = 12 m \]
We can compare the initial force \( F \) and the new force \( F' \) using the distances:
\[ F = G \frac{m_1 m_2}{(15)^2} \] \[ F' = G \frac{m_1 m_2}{(12)^2} \]
To find the factor by which the force changes, we can take the ratio of the new force to the initial force:
\[ \frac{F'}{F} = \frac{G \frac{m_1 m_2}{(12)^2}}{G \frac{m_1 m_2}{(15)^2}} = \frac{(15)^2}{(12)^2} \]
Calculating this gives:
\[ \frac{F'}{F} = \frac{225}{144} = \frac{225 \div 9}{144 \div 9} = \frac{25}{16} \]
This means that the gravitational force increases by a factor of:
\[ \frac{F'}{F} = \frac{25}{16} = 1.5625 \]
To express this in simpler terms, this represents an increase by a factor of \( \frac{25}{16} \), which can also be converted to a decimal or kept as a fraction indicating the factor by which the force increases.
Thus, the answer is:
- It increases by a factor of \( \frac{25}{16} \) or approximately \( 1.5625 \).
However, since \( 25/16 \) is not one of the options provided, we can find the closest relation to the answer choices.
If we analyze the options available:
- It increases by a factor of \( 1\frac{1}{4} = \frac{5}{4} \) (not applicable).
- It decreases by a factor of ( 1\frac{1}{4} = \frac{5}{4} (not applicable, but this indicates a decrease).
- It increases by a factor of ( 1\frac{9}{16} (not applicable).
- It decreases by a factor of ( 1\frac{9}{16} (not applicable, but indicates a decrease).
The gravitational force actually increases, but it doesn't fit the exact fractions. Given the calculations, you can confidently conclude that the gravitational force increases, but among the options provided, none directly represent the correct increase. Clarification of the correct results or reframing of the problem statement might be needed.