Equilibrium of Floating Cylinder

Given:
- Radius of cylinder, r = 3.0 m
- Height of cylinder, h = 9.0 m
- Specific gravity of cylinder material, SG = 0.85
- Cylinder is floating in water with its axis vertical

To determine whether the equilibrium is stable or unstable, we need to analyze the stability of the floating cylinder.

1. Buoyant Force:
The buoyant force acting on the cylinder is equal to the weight of the water displaced by the submerged volume of the cylinder. The buoyant force is given by the formula:

Buoyant force = weight of water displaced = ρgV

Where:
- ρ = density of water
- g = acceleration due to gravity
- V = volume of the submerged part of the cylinder

2. Weight of the Cylinder:
The weight of the cylinder is given by the formula:

Weight of cylinder = ρ_mgV_c

Where:
- ρ_m = density of cylinder material
- V_c = volume of the entire cylinder

3. Equilibrium Condition:
For the cylinder to be in equilibrium, the buoyant force and the weight of the cylinder should be equal. Mathematically,

Buoyant force = Weight of cylinder

ρgV = ρ_mgV_c

4. Volume of the Submerged Part:
The volume of the submerged part of the cylinder can be calculated as:

V = πr^2h

5. Volume of the Entire Cylinder:
The volume of the entire cylinder can be calculated as:

V_c = πr^2h_c

Where:
- h_c = height of the cylinder

6. Specific Gravity:
The specific gravity is the ratio of the density of the cylinder material to the density of water. Mathematically,

SG = ρ_m / ρ

7. Substituting the Equations:
Substituting the values of V, V_c, and SG in the equilibrium condition equation, we get:

ρgπr^2h = ρ_mgπr^2h_c

Simplifying the equation, we find:

h = SG * h_c

8. Equilibrium Analysis:
From the equation obtained in step 7, we can observe that the height of the submerged part of the cylinder is directly proportional to the height of the entire cylinder. Since the specific gravity of the cylinder material is less than 1 (0.85), the height of the submerged part (h) is less than the height of the entire cylinder (h_c).

Therefore, when the cylinder is slightly tilted, the center of buoyancy will shift towards the side of the cylinder with a larger height. This will create a restoring torque that brings the cylinder back to its original position. Hence, the equilibrium is stable.

Therefore, the correct answer is option 'A': Stable.