A thin uniform cylindrical shell, closed at both ends, is partially filed wih water. If pc is the relative density of the material of the shell with respect to water and it is floating half submerged in water, then the correct statement is that the shell is:-

1. More than half filled if pc
2. less than half filled if pc>1
3. more than half filled if pc>0.5
4. less than half filled if pc

This question was asked in IIT JEE ADVANCED 2012 PAPER 2 PHYSICS

To understand the relationship between the floating behavior of a thin uniform cylindrical shell partially filled with water and its relative density, let's break down the concepts step by step. When an object floats, the weight of the fluid displaced by the object is equal to the weight of the object itself. This principle is known as Archimedes' principle. In this case, we have a cylindrical shell that is closed at both ends and partially filled with water, floating half submerged in water.

Density and Buoyancy

The density of the material of the shell is given as a relative density (pc) with respect to water. Relative density is a dimensionless quantity defined as the ratio of the density of a substance to the density of water. If pc is less than 1, it indicates that the shell is less dense than water, while if pc is greater than 1, it shows that the shell is denser than water.

Analyzing Different Cases

Let's analyze the behavior of the shell for various ranges of relative density pc:

• If pc < 0.5: In this case, the shell is lighter than water and would float more than half submerged. This is because the buoyant force acting on the shell is greater than the weight of the shell itself, allowing it to displace more water.
• If pc = 0.5: The shell is exactly half submerged in water, which aligns with the condition given in the problem statement.
• If pc > 0.5 but < 1: The shell is still less dense than water, hence it will float less than half submerged. The buoyant force is not sufficient to support more than half of its volume being submerged.
• If pc = 1: The shell will be floating exactly at the water level, which is a special case of being half submerged.
• If pc > 1: Here, the shell is denser than water, which causes it to sink. Therefore, it cannot be floating and thus does not meet the floating condition described.

Conclusion on Floating Conditions

Based on the above analysis, we can summarize the statements regarding how the shell behaves based on its relative density:

• More than half filled if pc < 0.5
• Less than half filled if 0.5 < pc < 1
• Not floating (sinking) if pc > 1

So, the correct conclusion is that the cylindrical shell is more than half filled if pc < 0.5, and less than half filled if pc > 0.5. Understanding these principles helps clarify how buoyancy and density interact in fluid mechanics.