A thin uniform cylindrical shell, closed at both ends, is partially filled with water. It is floatin

Q: Why does gravity cancel out of the equilibrium equation?

Both the gravitational force (weight) and the buoyant force are directly proportional to the acceleration due to gravity, $g$. Since $g$ acts equally on all masses in the system, it cancels out, meaning the floating state depends purely on the balance of densities and volumes.

Q: What does relative density mean in this context?

Relative density $\rho_c$ is the ratio of the density of the shell material to the density of water. Since we take the density of water as $1\text{ g/cm}^3$, the relative density is numerically equal to the density of the material in these units.

Q: What happens if the relative density of the shell material is exactly 0.5?

If $\rho_c = 0.5$, the term $(0.5 - \rho_c)$ becomes zero. In this case, the fraction $x$ is exactly $0.5$, meaning the cavity must be exactly half-filled with water to keep the cylinder half-submerged.

Q: Why is the submerged volume taken as $(V_1 + V_2)/2$?

The total external volume of the closed cylinder is the sum of the material volume $V_1$ and the internal cavity volume $V_2$. Since the problem states that the cylinder is half-submerged, exactly half of this total external volume is underwater.

Q: What if the relative density of the material is greater than 0.5?

If $\rho_c > 0.5$, the term $(0.5 - \rho_c)$ becomes negative, which means $x < 0.5$. The cylinder would need to be less than half-filled with water to maintain its half-submerged state because the heavier material itself contributes more to the downward weight.

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

Options:

(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

Sigma Percentile · Accepted Answer

### The Magic of Flotation

Imagine holding a hollow, sealed cylinder in your hands. It feels light, but as you add water to its internal cavity, it grows heavier. If you place it in a pool, it floats vertically.

But here is the puzzle: what determines how much water we must add to keep it exactly **half-submerged**?

This is not just a question of weight; it is a beautiful dance between the density of the cylinder's material, the volume of its cavity, and the laws of buoyancy discovered by Archimedes over two thousand years ago.

Let's dive deep into the physics of this system and discover the elegant mathematical threshold that governs its behavior.

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### Setting Up the Balance

For any object floating in static equilibrium, the forces acting on it must perfectly balance out.

There are only two forces at play here:
1. The downward **gravitational force** (the total weight of the cylinder and the water inside it).
2. The upward **buoyant force** (the upthrust exerted by the displaced water).

Let's write down the mathematical expression for this equilibrium:

$$	ext{Total Weight} = 	ext{Upthrust}$$

Let's define our variables clearly:
* Let $V_1$ be the volume of the solid material making up the cylindrical shell.
* Let $ho_c$ be the relative density of this material with respect to water.
* Let $V_2$ be the volume of the internal cavity.
* Let $x$ be the fraction of the cavity volume filled with water.

Since the density of water is our reference unit ($ho_w = 1$), the total weight of the system is the sum of the weight of the shell and the weight of the water inside:

$$W_{	ext{total}} = [V_1 ho_c + x V_2] g$$

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### The Buoyant Force and Archimedes' Principle

According to Archimedes' Principle, the buoyant force is equal to the weight of the water displaced by the submerged portion of the cylinder.

Since the cylinder is floating in a **half-submerged** state, the volume of the displaced water is exactly half of the total external volume of the cylinder:

$$V_{	ext{submerged}} = \frac{V_{	ext{total}}}{2} = \frac{V_1 + V_2}{2}$$

Therefore, the upward buoyant force is:

$$F_B = \left(\frac{V_1 + V_2}{2}ight) g$$

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### The Mathematical Breakthrough

Now, we equate the downward weight to the upward buoyant force:

$$V_1 ho_c g + x V_2 g = \left(\frac{V_1 + V_2}{2}ight) g$$

Notice how the acceleration due to gravity, $g$, cancels out beautifully from both sides! This leaves us with a pure relationship of volumes and densities:

$$V_1 ho_c + x V_2 = \frac{V_1}{2} + \frac{V_2}{2}$$

To find the fraction $x$, we isolate the term containing $x$:

$$x V_2 = \frac{V_2}{2} + \left(\frac{1}{2} - ho_cight) V_1$$

Dividing the entire equation by $V_2$, we obtain our master equation:

$$x = 0.5 + (0.5 - ho_c) \frac{V_1}{V_2}$$

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### Analyzing the Density Threshold

This elegant equation tells a fascinating story. The fraction of the cavity filled with water, $x$, is equal to $0.5$ plus a correction term that depends entirely on whether the relative density of the material, $ho_c$, is greater or less than $0.5$.

Let's analyze the case where **$ho_c < 0.5$**:

* If $ho_c < 0.5$, then the term $(0.5 - ho_c)$ is strictly **positive**.
* Since the volume ratio $\frac{V_1}{V_2}$ is always positive, the entire correction term $(0.5 - ho_c) \frac{V_1}{V_2}$ is positive.
* Therefore, $x$ must be strictly greater than $0.5$:

$$x > 0.5$$

This means that if the material of the shell is very light (relative density less than $0.5$), the cylinder must be **more than half-filled** with water to overcome its natural buoyancy and remain half-submerged!

This perfectly matches **Option (a)**, making it the correct statement.

Select Answer:

Visualized Solution

Visualizing the Floating Cylinder

Archimedes' Principle & Equilibrium

Expressing the Total Weight

Expressing the Upward Buoyant Force

Setting up the Equilibrium Equation

Isolating the Fraction $x$

The Master Equation for $x$

Evaluating the Condition for Option (a)

Exploring Alternative Scenarios

The Sigma Insight: Buoyancy and Archimedes' Principle

The Magic of Flotation

Setting Up the Balance

The Buoyant Force and Archimedes' Principle

The Mathematical Breakthrough

Analyzing the Density Threshold

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