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Buoyancy is the force exerted by a fluid that opposes the weight of an object immersed in it. It's the reason why objects such as boats and barges can float. According to Archimedes' principle, the buoyant force on a submerged object is equal to the weight of the fluid it displaces.

For example, in the problem where a load of scrap metal is placed onto a barge, the barge sinks further into the water, displacing more water and thus experiencing a greater buoyant force. This increase in buoyant force continues until it balances the increased weight of the barge with the metal load. If the buoyant force is less than the weight of the object, the object will sink until it rests on the bottom or is fully submerged.

Density is a property that measures how much mass is contained in a given volume. It's mathematically defined as mass per unit volume and typically expressed in units such as kilograms per cubic meter (\text{kg/m}^{3}).

The concept of density is crucial in understanding why certain objects float or sink. For instance, the problem's reference to the density of the metal at 7200 \(\mathrm{kg/m}^{3}\) is much greater than the density of freshwater, which is about 1000 \(\mathrm{kg/m}^{3}\). An object will sink if its density is greater than the fluid it displaces, which is exactly what happens to the metal when pushed overboard into the water.

Displacement in physics usually refers to the volume of fluid that an object pushes aside when it is placed in the fluid. This concept is directly related to the calculation of buoyancy. The displaced fluid's volume is equivalent to the volume of the part of the object that is submerged.

In the situation where the scrap metal is added to the barge, it displaces an amount of water equal to the volume of metal. If the metal is later pushed overboard, it will displace an amount of water equivalent to its own submerged volume, which affects the water level in the lock. Displacement helps us understand changes in water level, as seen in this problem.

Fluid mechanics is the study of fluids (liquids, gases, and plasmas) and the forces on them. It's divided into fluid statics (studying fluids at rest) and fluid dynamics (studying fluids in motion).

This area of physics provides the theoretical foundation for understanding the problem's scenario, where the barge's interaction with the water is a fluid statics problem. Fundamental principles from fluid mechanics, such as Archimedes’ principle and the concept of pressure, directly contribute to solving problems involving buoyancy, density, and displacement.