A thin uniform cylindrical shell , closed at both ends it is partially filled with water .It is a floating in half submerged state .If Pc is the relative density of the material of the shell .what is correct statement

To analyze the situation of a thin uniform cylindrical shell that is partially filled with water and floating in a half-submerged state, we need to consider the principles of buoyancy and density. The shell's behavior in water is governed by Archimedes' principle, which states that the buoyant force acting on a submerged object is equal to the weight of the fluid displaced by that object.

Understanding the Forces at Play

When the cylindrical shell is floating, it displaces a volume of water equal to the weight of the shell plus the weight of the water inside it. The key factors to consider here are:

• Weight of the Shell: This is determined by the material density (relative density, Pc) and the volume of the shell.
• Weight of the Water Inside: This is based on the volume of water inside the shell and the density of water.
• Buoyant Force: This is equal to the weight of the water displaced by the submerged part of the shell.

Applying Archimedes' Principle

For the shell to float in a stable equilibrium, the buoyant force must equal the total weight of the shell and the water it contains. Mathematically, this can be expressed as:

Buoyant Force = Weight of the Shell + Weight of the Water

Let’s denote:

• V_shell = Volume of the shell
• V_water = Volume of water inside the shell
• ρ_shell = Density of the shell material
• ρ_water = Density of water (approximately 1000 kg/m³)

The weight of the shell can be expressed as:

Weight_shell = ρ_shell × V_shell × g

The weight of the water inside the shell is:

Weight_water = ρ_water × V_water × g

The buoyant force is given by the weight of the displaced water, which can be calculated as:

Buoyant Force = ρ_water × V_displaced × g

Finding the Relationship

In a half-submerged state, the volume of water displaced (V_displaced) is equal to half the volume of the shell (assuming the shell is symmetrical and uniformly filled). Therefore:

V_displaced = 0.5 × V_shell

Substituting this into the buoyant force equation gives:

Buoyant Force = ρ_water × (0.5 × V_shell) × g

Setting the buoyant force equal to the total weight of the shell and the water gives:

ρ_water × (0.5 × V_shell) × g = ρ_shell × V_shell × g + ρ_water × V_water × g

Relative Density Consideration

By simplifying this equation, we can derive a relationship involving the relative density of the shell material (Pc). The relative density is defined as:

Pc = ρ_shell / ρ_water

From the equilibrium condition, we can conclude that:

0.5 = Pc + (V_water / V_shell)

In this case, if the shell is floating in a half-submerged state, it implies that the average density of the system (shell + water) must be equal to the density of the water. Therefore, the correct statement regarding the relative density of the shell material is:

Final Statement

The relative density of the shell material (Pc) must be less than or equal to 0.5 for the shell to float in a half-submerged state. This indicates that the shell is less dense than the water, allowing it to float while partially filled.