Final answer:
The force of gravity between two objects is influenced by both their masses and the distance between them. Doubling the mass of objects quadruples the gravitational force, while tripling the separation distance decreases the force by a factor of nine, due to the inverse-square law.
Step-by-step explanation:
Gravity and Mass-Distance Relationship
The force of gravity between two objects is directly proportional to the product of their masses and inversely proportional to the square of the distance between their centers as described by Newton's law of universal gravitation. When the mass of each object is doubled, the gravitational force becomes four times greater because the product of their masses is quadrupled. However, when the distance between objects is tripled, the gravitational force is decreased by a factor of nine (3^2), in alignment with the inverse-square law.
Weight is defined as the gravitational force exerted on an object by a massive body such as Earth. Thus, if we double an object's mass, its weight also doubles, assuming the distance from the center of mass (Earth's core) remains constant. Conversely, if an object's distance from the center of Earth is doubled, its weight is reduced to one-fourth, whereas if the distance is tripled, the weight would then decrease to one-ninth.
In conclusion, the relationship between gravitational force, mass, and distance is crucial for understanding not only the orbits of planets but also the way objects experience gravity in our daily lives. This understanding helps explain phenomena such as the weight changes of objects at different altitudes above Earth's surface and is fundamentally based on Newton's work and calculus.