To solve the problem of the thin-walled cylinder floating on water and experiencing a change in submersion due to gas leakage, we need to analyze the forces acting on the cylinder and apply the principles of fluid mechanics. Let's break this down step by step.

Understanding the Forces at Play

The cylinder is initially floating, which means the buoyant force acting on it is equal to the weight of the cylinder plus the weight of the gas inside it. When the gas leaks, the weight of the gas decreases, causing the cylinder to submerge deeper into the water.

Key Variables

• m: mass of the cylinder
• h: initial height of the cylinder above the water surface
• A: cross-sectional area of the cylinder
• P₀: atmospheric pressure
• g: acceleration due to gravity
• h': increase in submersion depth (equal to h)

Applying Archimedes' Principle

According to Archimedes' principle, the buoyant force (B) acting on the cylinder is equal to the weight of the water displaced by the submerged part of the cylinder. The weight of the water displaced can be calculated as:

B = ρ_water * g * V_displaced

Where:

• ρ_water is the density of water
• V_displaced is the volume of water displaced, which can be expressed as A * (h + h')

Weight of the Cylinder and Gas

The total weight (W) of the cylinder and the gas inside it is given by:

W = m * g + P_initial * V_gas

Where V_gas is the volume of the gas inside the cylinder, which can be expressed as A * h.

Setting Up the Equation

Initially, when the cylinder is floating, the buoyant force equals the total weight:

ρ_water * g * A * h = m * g + P_initial * A * h

After Leakage

After the gas leaks, the new depth of submersion is h + h, and the new buoyant force becomes:

B' = ρ_water * g * A * (h + h)

Setting this equal to the new weight of the cylinder and gas gives us:

ρ_water * g * A * (h + h) = m * g + P_initial' * A * (h + h)

Finding the Initial Pressure

We can rearrange the equation to isolate the initial pressure of the gas:

P_initial = P₀ + (m * g) / A - (h / h) * P_initial'

Since the temperature remains constant, we can assume that the pressure of the gas inside the cylinder is directly related to the volume it occupies. Therefore, we can express the initial pressure of the gas as:

P_initial = P₀ + (m * g) / A

Final Expression

Thus, the initial pressure of the gas in the cylinder can be summarized as:

P_initial = P₀ + (m * g) / A - (h / h)

This equation shows how the initial pressure of the gas is influenced by the atmospheric pressure, the mass of the cylinder, and its cross-sectional area. It highlights the balance between the buoyant force and the weight of the cylinder and gas, which is crucial for understanding the dynamics of floating objects in fluids.