The correct answer is Option 3: Archimedes’ principle.

Explanation

Archimedes' principle states that when a body is immersed fully or partially in a fluid, it experiences an upward buoyant force that is equal to the weight of the fluid displaced by the body.

Here is how it applies to the items listed:

Ships and Submarines: A ship floats because its hollow shape displaces a volume of water whose weight is equal to or greater than the ship's own weight. Submarines use ballast tanks to take in or release water, changing their average density and weight to submerge or rise based on this principle.

Hydrometers: These instruments measure the density (or specific gravity) of liquids. They sink deeper in less dense liquids and float higher in denser liquids because the weight of the displaced liquid must always equal the weight of the hydrometer.

Lactometers: A specialized type of hydrometer used to test the purity of milk by measuring its density based on Archimedes' principle.

Why the Other Options Are Incorrect

Option 1 (Principle of gravitation): While gravity pulls the objects downward, it does not explain how they float, sink, or measure fluid density.

Option 2 (Principle of relative density): While hydrometers and lactometers measure relative density, they are designed and operate based on the mechanical law of Archimedes' principle.

Option 4 (Principle of buoyancy): Buoyancy is the upward force itself, but Archimedes' principle is the definitive scientific law that quantifies this force and dictates the design framework for these specific instruments and vessels.