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Specific gravity is a crucial concept when discussing buoyancy and floating objects. It is defined as the ratio of the density of a substance to the density of a reference substance, typically, water. This ratio allows us to understand how heavy or light a substance is concerning water. Specific gravity is a dimensionless quantity because it is a ratio and therefore has no units. This property makes it quite handy in comparing the buoyancy of various materials without the need to focus on specific units.

• If the specific gravity is less than 1, the substance is lighter than water and will float.
• If it is exactly 1, the substance has the same density as water.
• If greater than 1, the substance is denser and will sink.

In the problem, understanding specific gravity helps determine how the liquid added affects the buoyancy of the container. By calculating its value, you can assess the contribution of the liquid's weight to the immersed portion of the container.

Equilibrium in fluids occurs when the forces acting upon an object within a fluid are balanced. This typically involves the buoyant force and the gravitational force. An object floats when the buoyant force equals the gravitational force acting on it, often called "weight." In this state of equilibrium, the object neither sinks nor rises but stays at a consistent level within the fluid.
For a floating object like the cylindrical container in the problem, equilibrium is indicated by the level to which the container is submerged. Initially, the container floats stably with half of its length submerged, showing that the forces are balanced. The addition of liquid changes this balance, leading to a new state where the container is submerged to two-thirds of its length.
This adjustment showcases equilibrium, as the system reaches a new balance between the container's weight and the buoyant force. Observing how a system maintains or shifts equilibrium offers insight into fluid interactions and is a fundamental aspect of fluid dynamics.

Archimedes' Principle is the cornerstone of understanding buoyancy. It states that any object, wholly or partially submerged in a fluid, experiences a buoyant force equal to the weight of the fluid displaced by the object. This principle explains why objects sink, float, or remain neutrally buoyant in fluid environments.

• When an object is fully submerged, the displaced fluid's weight equals the object's weight, causing it to float at a constant depth.
• When partially submerged, the buoyant force corresponds to the volume of fluid displaced, which must balance the object's weight for it to float stably at the surface.

In the exercise, Archimedes' Principle helps us understand why the cylinder floats with half of its length submerged initially. As liquid is added, the system transitions to a new floating state where Archimedes' Principle still governs the relationship between displaced water weight and the cylinder's overall weight.
This principle provides a systematic method to address varying fluid mechanics problems involving floating or submerged objects, as seen when calculating the required specific gravity to achieve equilibrium after adding liquid to the container.