If the moon moved closer to the earth, The gravitational force between them will increase.
This can be understood by Newton's law of universal gravitation. This is usually stated as that every particle attracts every other particle in the universe with a force that is directly proportional to the product of their masses and inversely proportional to the square of the distance between their centers.
[tex]F_{gravitation}[/tex] = G[tex]\frac{M_{1} M_{2}}{r^{2} }[/tex]
If the moon moved closer to earth, the distance between them will be reduced, and as the force of gravitation is inversely proportional to the distance between the masses, the gravitation force increases.
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