Cylindrical Submarine and Metal Plate Thickness Calculation

To determine the thickness of the metal plate used in the fabrication of the Arihant class submarine, we need to consider the hydrostatic forces acting on the submarine when it is half submerged in sea water.

Hydrostatic Forces on the Submarine

When the submarine is half submerged, it experiences buoyancy forces due to the weight of the water displaced by the submerged portion. To calculate the thickness of the metal plate, we need to determine the hydrostatic pressure acting on the submarine.

The hydrostatic pressure is given by the equation:
P = ρ * g * h

Where:
P is the hydrostatic pressure
ρ is the density of the sea water
g is the acceleration due to gravity
h is the depth of the submerged portion of the submarine

Density of Sea Water

Given the specific gravity (SG) of the sea water as 1.05, we can calculate the density (ρ) using the equation:
ρ = SG * ρ_water

Where:
ρ_water is the density of fresh water (1000 kg/m³)

So, ρ = 1.05 * 1000 = 1050 kg/m³

Depth of the Submerged Portion

The depth of the submerged portion of the submarine can be determined using the dimensions of the submarine. Since the submarine is half submerged, the depth can be calculated as:
Depth = (15/2) - (15/2) * (1/2) = 15/4 = 3.75 m

Hydrostatic Pressure Calculation

Substituting the values of ρ, g, and h into the hydrostatic pressure equation, we get:
P = 1050 * 9.81 * 3.75 = 38,682.375 Pa

Thickness Calculation

The thickness of the metal plate can be determined by considering the hoop stress acting on the submarine. The hoop stress is given by the equation:
Stress = P * r / t

Where:
Stress is the hoop stress
P is the hydrostatic pressure
r is the radius of the submarine
t is the thickness of the metal plate

We can rearrange the equation to solve for t:
t = P * r / Stress

Submarine Radius Calculation

Given the diameter of the submarine as 15 m, the radius (r) can be calculated as:
r = 15 / 2 = 7.5 m

Assuming a Safe Stress Value

The stress value depends on the material properties of the metal plate. Assuming a safe stress value of 200 MPa (MegaPascal), we can calculate the thickness of the metal plate:
t = 38,682.375 * 7.5 / 200 * 10⁶ = 0.145 m = 145 mm

Therefore, the thickness of the metal plate used in the fabrication of the Arihant class submarine is 145 mm.

Volume of Water for Complete Submersion

To determine the volume of water required for complete submersion of the submarine, we need to consider the submerged volume of the submarine.

The submerged volume of a cylinder can be calculated using the formula:
V = π * r² * h

Where: