To solve this problem, we need to analyze the forces acting on the floating cylinder and apply the principles of fluid mechanics and gas laws. The situation involves a thin-walled cylinder that is partially submerged in water, and as gas leaks from it, the depth of submergence increases. Let's break this down step by step.

Understanding the Forces at Play

Initially, the cylinder is floating, which means the buoyant force acting on it is equal to its weight. The buoyant force can be calculated using Archimedes' principle, which states that the upward buoyant force is equal to the weight of the fluid displaced by the submerged part of the cylinder.

Initial Conditions

Let’s denote:

• m = mass of the cylinder
• h = initial height of the cylinder
• A = cross-sectional area of the cylinder
• p₀ = atmospheric pressure
• p₁ = initial pressure of the gas inside the cylinder
• g = acceleration due to gravity

The weight of the cylinder is given by:

Weight = mg

Buoyant Force Calculation

The initial buoyant force when the cylinder is floating is:

Buoyant Force = ρ_wgV_sub

where ρ_w is the density of water and V_sub is the volume of the submerged part of the cylinder. Initially, if the cylinder is submerged to a depth d, then:

V_sub = A * d

Setting the buoyant force equal to the weight of the cylinder gives us:

ρ_wgA * d = mg

From this, we can derive the initial depth of submergence:

d = (m / ρ_wA)

After Leakage Occurs

As gas leaks from the cylinder, the depth of submergence increases by h, making the new submerged depth d + h. The new buoyant force can be expressed as:

Buoyant Force_new = ρ_wgA(d + h)

Setting this equal to the weight of the cylinder, we have:

ρ_wgA(d + h) = mg

Relating Pressures

Since the temperature remains constant, we can apply Boyle's Law, which states that for a given mass of gas at constant temperature, the pressure and volume are inversely related:

p₁V₁ = p₂V₂

Initially, the volume of gas in the cylinder is:

V₁ = A * (h - d)

After the leakage, the volume of gas becomes:

V₂ = A * (h - (d + h)) = A * (h - d - h) = A * (-d)

However, since the cylinder is floating, we can express the pressures in terms of the atmospheric pressure:

p₁ = p₀ + ρ_wgh

And after the leakage, the pressure becomes:

p₂ = p₀ + ρ_wg(d + h)

Final Equation

By applying Boyle's Law:

(p₀ + ρ_wgh)(A(h - d)) = (p₀ + ρ_wg(d + h))(-d)

From this equation, we can isolate p₁ and solve for it. The calculations will yield the initial pressure of the gas in the cylinder.

In summary, the initial pressure of the gas can be determined by balancing the forces and applying the gas laws, which ultimately leads us to understand how the changes in volume and pressure relate to the buoyancy of the cylinder in water.