Question

A thin uniform cylindrical shell, closed at both ends, is partially filled with water. It is floating vertically in water in half-submerged state. If \(\rho_{c}\) is the relative density of the material of the shell with respect to water, then the correct statement is that the shell is (A) more than half-filled if \(\rho_{c}\) is less than \(0.5\). (B) more than half-filled if \(\rho_{c}\) is more than \(1.0 .\) (C) half-filled if \(\rho_{c}\) is more than \(0.5\). (D) less than half-filled if \(\rho_{c}\) is less than \(0.5\).

Short answer

(A) more than half-filled if \(\rho_c\) is less than 0.5.

Step-by-step solution

Understand Buoyancy

Recall that the buoyant force on a submerged object is equal to the weight of the displaced fluid. For the cylindrical shell floating in water, the buoyant force needs to balance the weight of the shell itself as well as the weight of the water it contains. Since it is floating, the volume of water displaced by the submerged part of the shell is equal to the volume of the entire shell.

Relate Buoyancy to Cylinder Fill Level

Let the total volume of the cylinder be V. If the cylinder is floating vertically and half-submerged, it displaces water volume V/2. The weight of the displaced water is equal to the weight of the shell plus the weight of the water inside. The volume of water inside the cylinder that makes it half-submerged is dependent on the relative density of the shell material.

Write the Equation for Equilibrium

Let V_w be the volume of the water inside the shell. The force due to the weight of the displaced water needs to be equal to the force due to the weight of the water inside the shell (V_w\(\rho_{water}g\)) and the weight of the shell material (V\(\rho_c\rho_{water}g\)). This results in the equation \(V_w \rho_{water}g + V \rho_c \rho_{water}g = \frac{V}{2} \rho_{water}g\).

Solve for the Volume of Water Inside the Shell

Simplifying the equilibrium equation gives \(V_w + V \rho_c = \frac{V}{2}\). Solve for the volume of water inside the shell: \(V_w = \frac{V}{2} - V \rho_c\).

Analyze the Relationship Between V_w and V

By analyzing the equation \(V_w = \frac{V}{2} - V \rho_c\), you can see that when \(\rho_c < 0.5\), the volume V_w is more than V/2 (making the shell more than half-filled), and when \(\rho_c > 0.5\), V_w is less than V/2 (making the shell less than half-filled).

Choose the Correct Statement

Based on the relationship, it can be concluded that if the relative density \(\rho_c\) is less than 0.5, the shell is more than half-filled with water to float half-submerged. This corresponds to option (A).

Key concepts

Buoyant Force

Imagine submerging an object in water and feeling it push back against you. This push is the buoyant force, a special phenomenon where a fluid (like water) exerts an upward force on any object placed in it. According to Archimedes' principle, this force is equal to the weight of the fluid displaced by the object. It's like the fluid is trying to make room for the object by pushing it up.

Think of a boat floating on water. The bottom part pushes water aside, and the water pushes back, keeping the boat afloat. In the given exercise, the cylindrical shell is the 'boat,' and the water inside and outside are together working to balance the shell upright in the water.

Relative Density

The term relative density, or specific gravity, refers to how dense one material is in relation to another, usually water. It's a ratio without units, comparing the density of a substance to the density of water. If the relative density is greater than 1, the substance is denser than water and will likely sink. If it's less than 1, it's less dense and will tend to float.

In our exercise, the relative density \(\rho_{c}\) of the cylindrical shell material compared to water is the key player in determining whether the cylinder is more or less than half-filled with water to maintain a half-submerged equilibrium.

Floating Equilibrium

For an object to achieve a floating equilibrium, the forces at play must be perfectly balanced. It means that the upward buoyant force exerted by the fluid must match the downward gravitational force of the object's weight. If these forces didn't balance out, the object would either sink or float entirely on the surface.

In floating equilibrium, like our half-submerged cylindrical shell, the volume of fluid displaced (which dictates the buoyant force) is related to the weight of the object—and the result is a delicate balance that keeps the cylinder stable in its half-submerged state.

Submerged Volume

The submerged volume is quite literally the portion of an object that's underwater. For an object floating in water, only part of it is submerged, and this part displaces a certain volume of water. The weight of this displaced water is significant because it is precisely equal to the buoyant force acting upon the object.

In the context of our cylinder, its half-submerged state indicates that the submerged volume is equal to half the total volume of the cylinder. This volume is crucial because it tells us how much water has been displaced, which in turn tells us about the buoyant force keeping the cylinder afloat.