System A has masses m and m separated by a distance r ;...

System A has masses $m$ and $m$ separated by a distance $r ;$ system $B$ has masses $m$ and 2$m$ separated by a distance $2 r ;$ system $C$ has masses 2$m$ and 3$m$ separated by a distance $2 r,$ and system $D$ has masses 4 $\mathrm{m}$ and 5 $\mathrm{m}$ separated by a distance 3$r .$ Rank these systems in order of increasing gravitational force. Indicate ties where appropriate.

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Key Concepts

Proportional Reasoning for Comparison

When specific numerical constants are not provided, one can rank or compare systems based on the relative sizes of the mass products and the squared distances. This reasoning allows for a systematic comparison of gravitational forces across different systems.

Inverse Square Law

The force due to gravity decreases with the square of the distance between the two masses. This means that as the separation between masses increases, the gravitational force diminishes rapidly, which is critical when evaluating systems with different distances.

Newton's Law of Gravitation

This law explains that the gravitational force between two masses is proportional to the product of the masses and inversely proportional to the square of the distance between them. It is essential in analyzing any problem involving gravitational attraction between two bodies.

Mass Product Effect

The gravitational force is influenced directly by the product of the two interacting masses. Larger mass products lead to a stronger gravitational pull if the distance between the masses remains constant, making it a key factor in comparisons.

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