Problem Statement: A cylinder of diameter 300 mm having specific gravity 0.8 floats in water. What is the maximum permissible length in order the cylinder may floats in stable equilibrium with its axis vertical? Explain in details.

Solution:

Given:

Diameter of cylinder, D = 300 mm

Specific gravity of cylinder, S.G. = 0.8

Density of water, ρ = 1000 kg/m³

Acceleration due to gravity, g = 9.81 m/s²

Assumptions:

The cylinder is considered to be a circular cylinder.

The cylinder is completely submerged in water.

The cylinder is in stable equilibrium.

Formula:

The formula to calculate the maximum permissible length of the cylinder is given by:

L = (ρ/2S.G.) × D

Where,

L = Maximum permissible length of the cylinder

D = Diameter of cylinder

S.G. = Specific gravity of cylinder

ρ = Density of water

Calculation:

Substituting the given values in the above formula, we get:

L = (ρ/2S.G.) × D

L = (1000/2 × 0.8) × 0.3

L = 187.5 mm

Therefore, the maximum permissible length of the cylinder is 187.5 mm in order to float in stable equilibrium with its axis vertical.

Explanation:

When a cylinder is partially or completely submerged in a fluid, it experiences an upward force called buoyancy. This buoyancy force is equal to the weight of the fluid displaced by the cylinder. If the weight of the cylinder is less than the buoyancy force, the cylinder will float in the fluid.

The specific gravity of the cylinder is given as 0.8, which means that the cylinder is lighter than water. Thus, the cylinder will float in water.

To determine the maximum permissible length of the cylinder, we can use the formula mentioned above. This formula relates the maximum permissible length of the cylinder with its diameter, specific gravity, and the density of the fluid in which it is submerged.

In this case, we substitute the given values of diameter, specific gravity, and density of water in the formula and calculate the maximum permissible length of the cylinder.

Hence, we can conclude that the maximum permissible length of the cylinder is 187.5 mm in order to float in stable equilibrium with its axis vertical.