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Density is a fundamental concept in understanding buoyancy and floating objects. It is defined as the amount of mass contained in a unit volume of an object or substance. Mathematically, density is expressed as:\[\text{Density} = \frac{\text{mass}}{\text{volume}}\]For example, if an object has a mass of 250 kg and occupies a volume of 1 cubic meter, its density will be 250 kg/m³. The concept of density helps us determine whether an object will float or sink when placed in a fluid. Objects with a lower density than the fluid will float, while those with a higher density will sink. In our exercise, the density of the duck is crucial because it must be less than the density of water (1000 kg/m³) for it to float, which it indeed is, as calculated to be 250 kg/m³.

Displacement is a key concept when discussing objects immersed in fluids. It refers to the volume of fluid that is moved out of the way or displaced by an immersed object. When a duck floats on water, it pushes aside a certain amount of water equal to the volume submerged, in this case, 25% of its total volume. Displacement volume plays a critical role in determining buoyancy, as the weight of the displaced fluid is used to calculate the buoyant force acting on the object. This buoyant force is what counterbalances the weight of the duck, allowing it to float. Understanding displacement is essential to apply Archimedes' principle and solve related physics problems effectively.

Archimedes' principle is a cornerstone of fluid mechanics and explains why objects float or sink. It states that any object, fully or partially submerged in a fluid, experiences a buoyant force equal to the weight of the fluid that the object displaces. This principle lays the foundation for understanding buoyancy. For the floating duck, the buoyant force is equal to the weight of 25% of the duck's volume in water. This means even though only a quarter of the duck's volume is below water, it is enough to keep it afloat since the weight of displaced water equals the weight of the entire duck. Archimedes' principle is applied in calculating buoyancy, allowing us to find the density of the duck using known properties of water.

Floating objects are those that remain on the surface of a fluid without sinking. The ability of an object to float depends primarily on the relationship between the object's density and the fluid's density. If the object has a density lower than the fluid, it will generally float. The duck in our problem floats because its average density (250 kg/m³) is lower than that of water (1000 kg/m³). Only 25% of the duck's volume needs to displace water to match its weight, thus maintaining buoyancy. This allows large portions of the duck to sit above the water's surface. Studying floating objects helps in understanding natural phenomena and designing ships, boats, and other floating structures safely.