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Fluid mechanics is the branch of physics that studies the behavior of fluids (liquids and gases) and the forces acting upon them. Understanding fluid mechanics is essential to solve problems involving floating objects, like our hollow cubical box. When the box is floating in the lake, it interacts with the water around it. The water exerts an upward force on the box called the buoyant force. At the same time, gravity pulls the box downward. These two forces need to balance each other for the box to float stably without sinking or rising. Fluid mechanics helps us analyze and predict how fluids move and the forces they exert. It guides us to calculate whether an object will float or sink under specific conditions, like when water is added to the box in our scenario.

Archimedes' Principle is a fundamental principle in fluid mechanics. It states that any object fully or partially submerged in a fluid is buoyed up by a force equal to the weight of the fluid displaced by the object. In simple terms, the principle explains why things float. For our cubical box, one-third of its height is beneath the water surface. According to Archimedes' Principle, the box displaces an amount of water with a weight equal to the weight of the box. This balance of forces allows the box to float. When water is poured into the box, the total displaced volume must rise to match the weight of the box plus the additional water. This is why when the internal water depth reaches a certain level, it leads to a new equilibrium between displaced volume and gravitational force.

Displacement volume has a crucial role in floating objects and is directly tied to buoyancy. It refers to the volume of the fluid that is moved out of its original place when an object is submerged.For our cube, the initially displaced volume is \[0.10 imes 0.30 imes 0.30 = 0.009 \, \text{m}^3\] This is the volume of water equal to one-third of the cube's submerged height, explaining why it floats partially. As more water is added inside, the interior water increases displacement.When enough water is added to match the outside water level, the internal and external displaced volumes equate. This means the water depth in the box becomes equal to the submerged depth of the box, reaching 0.10 m. When displacement volume inside equals that of the outside, forces balance and water starts to flow in from the lake.