The Law of Universal Gravitation

Newton's Law of Universal Gravitation states that the force of gravity between two objects is:

* Directly proportional to the product of their masses: Larger masses exert a stronger gravitational pull.

* Inversely proportional to the square of the distance between their centers: As objects get closer, the gravitational force increases rapidly.

The Math

Let's break it down:

* Original force: Let's call the original force of gravity 'F'.

* Original distance: Let's call the original distance between the objects 'd'.

* New distance: We're moving the objects 3 times closer, so the new distance is 'd/3'.

Now, let's see how the force changes:

1. Original force formula: F = (G * m1 * m2) / d²

2. New force formula: F' = (G * m1 * m2) / (d/3)²

3. Simplifying: F' = (G * m1 * m2) / (d²/9) = 9 * (G * m1 * m2) / d²

4. Relationship: F' = 9F

Conclusion

By moving the objects 3 times closer, we've decreased the distance by a factor of 3. Because the force of gravity is inversely proportional to the *square* of the distance, the force increases by a factor of 3² = 9.