To tackle the problem of a thin uniform cylindrical shell that is partially filled with water and floating vertically, we need to consider the principles of buoyancy and the relationship between the density of the shell and the water. The key concept here is that for an object to float, the weight of the water displaced must equal the weight of the object itself, including any contents. Let's break this down step by step.
Understanding Buoyancy
According to Archimedes' principle, the buoyant force acting on a submerged object is equal to the weight of the fluid displaced by that object. In our case, the cylindrical shell is floating, which means it displaces a volume of water equal to its weight.
Density and Relative Density
The relative density (or specific gravity) of the shell material, denoted as ρC, is a comparison of the density of the shell material to that of water. If we denote the density of water as ρ_water (approximately 1 g/cm³), then:
- If ρC < 1, the shell is less dense than water.
- If ρC = 1, the shell has the same density as water.
- If ρC > 1, the shell is denser than water.
Analyzing the Floating Condition
When the shell is floating in a half-submerged state, it means that the volume of water displaced is equal to half the volume of the shell. The weight of the displaced water can be expressed as:
Weight of displaced water = Volume displaced × ρ_water × g
For the shell, the total weight can be expressed as:
Weight of shell = Volume of shell × ρC × g
Setting these two weights equal gives us:
Volume displaced × ρ_water = Volume of shell × ρC
Determining the Filling Condition
Now, let's analyze the implications of different values of ρC:
- If ρC < 0.5, the shell is very light compared to water. This means it would need to displace more water to balance its weight, indicating that it is more than half-filled.
- If ρC = 0.5, the shell is exactly balanced at half-filled.
- If ρC > 0.5 but < 1.0, the shell is less than half-filled because it is denser than the water it displaces.
- If ρC = 1.0, the shell is half-filled.
- If ρC > 1.0, the shell is denser than water, meaning it would be less than half-filled to float.
Conclusion on Options
Based on this analysis, we can evaluate the provided options:
- (A) More than half-filled if ρC is less than 0.5: This is true.
- (B) More than half-filled if ρC is more than 1.0: This is false; it would be less than half-filled.
- (C) Half-filled if ρC is more than 0.5: This is false; it would be less than half-filled if ρC > 1.0.
- (D) Less than half-filled if ρC is less than 0.5: This is false; it would be more than half-filled.
Thus, the correct statement is (A): the shell is more than half-filled if ρC is less than 0.5. This understanding of buoyancy and density relationships is crucial in solving problems related to floating objects.
