Basic Stability, Flotation Part-1

Density of a substance is its mass per unit volume.
Density = Mass/Volume
Where mass is in tonnes (t), Volume is in cubic metres (m^3), Density is in tonnes per cubic metre (tm^3)
Density of water is 1 tm^3
Relative density is the number of times a substance is heavier than water. Being a ratio, RD has no Unit.

•Vol. of tank = vol. of oil + vol. of free space

•Free Vol. =   l*b*h                                                =>     h = free vol. /  ( l*b )                                  where, h is ullage.

Pressure is the load per unit area.
Pressure = density * depth
unit = t/m2
1 bar = 10.2 t/m2
1 ton/m2 = 9.81 kN/m2

Thrust is the total pressure exerted on a given surface.
Thrust = pressure * area
unit => ton or kN
1 ton = 9.81 kN

Archimedes Principle states that when a body is totally or partially submerged in a fluid, it suffers an apparent loss of weight which is equal to the weight of fluid displaced.

Since the word fluid includes both, liquids and gases, and the fact that merchant ships are only expected to be partially immersed in water, a modified version of Archimedes' Principle may be called the Principle of flotation

Principle of flotation: When a body is floating in a liquid, the weight of liquid displaced equals to the weight of the body.
Displacement is commonly used to denote the mass of a ship in tonnes. Technically, it is the mass of water displaced by a ship and, when floating freely, the mass of water displaced equals to the mass of the ship.

Light displacement is the mass of the empty ship - without any cargo, fuel, lubricating oil, ballast water, fresh and feed water in tanks, consumable stores, and passengers and crew and their effects
Load displacement is the total mass of the ship when she is floating in salt water with her summer loadline at the water surface.
Present displacement is the mass of the shjp at present. It is the sum of the light displacement of the ship and everything on board at present.

Deadweight (DWT) of a ship is the total mass of cargo, fuel, freshwater, etc., that a ship can carry, when she is floating in salt water with her summer loadline at the water surface.
DWT = Load displacement – Light displacement
Deadweight aboard is the total mass of cargo, fuel, ballast, fresh water, etc., on beard at present.
DWT aboard = present displ -  light displ

Deadweight available is the total mass of cargo, fuel, fresh water, etc., that can be put on the ship at present to bring her summer loadline to the water surface in salt water.
DWT available = load displ - present displ

Waterplane coefficient (Cw), or coefficient of fineness of the water-plane area, is the ratio of the area of the water-plane to the area of a rectangle having the same length and maximum breadth.
Cw = Area of water-plane
              L x B

Area of water-plane = L x B x Cw

Block coefficient (Cb), or Coefficient of fineness of displacement, at any draft is the ratio of the underwater volume of the ship at that draft to a rectangular box having the same extreme dimensions.
Cb = Underwater volume
LxBxd

The term block coefficient may also be used with respect to a tank in which case it would be the ratio of the volume of the tank to the volume of a rectangular box having the same extreme dimensions as the tank
Cb  = Volume of tank
  LxBxD
Volume of tank = L x B x D x Cb

Reserve buoyancy (RB) is the volume of the enclosed spaces above the waterline. It maybe expressed as a volume in m3 or as a percentage of the total volume of the ship.
RB =  Total volume - underwater volume
RB % = Above water volume x 100
             Total volume

Reserve buoyancy is so called because, though it is not displacing any water at that time, it is available for displacement if weights are added or if bilging takes place. Bilging is the accidental entry of water into a compartment, due to underwater damage

Tonnes per centimetre (TPC) is the number of tonnes required to cause the ship to sink or rise by one centimetre

Considering 1 cm sinkage

Increase in underwater volume = A x  1/100 m^3
Increase in W = A/100 x density of water displaced.
Or TPC = A/100  x density of water displaced
TPC in SW = A/100 x 1.025  = 1.025A/100
TPC in FW = A/100
TPC in DW of density RD  =  (RD x A)/ 100

In the foregoing formulae, the area of the water-plane of a ship-shape has been considered constant since the sinkage or rise being considered is only 1 cm. However, the area of the water-plane of a ship-shape usually increases as draft increases. Hence, its TPC also increases as draft increases.

In view of this, calculations involving TPC should generally be confined to small values of sinkage or rise, say less than about 20 cm, in the case of ship-shapes. Otherwise, the accuracy of the calculation will tend to suffer.

In the case of a box-shaped vessel, the area of the water-plane is the same at all drafts and hence its TPC does not change with draft.

Page 24, Problem 3 of Stability book 1 by Capt. Subramaniam please refer there also.
Rectangular Log  B=3m, H=2M floats with breadth horizontal. Density of log is 0.7t/m3
Find its draft in water of  R.density 1.01
Vol of log = l x 3 x 2
Weight of log = l x 3 x 2 x 0.7
Being a homogenous uniform log, ratio of  weight/volume will be the ratio of immersion to the height in fresh water.
Therefore, draft in FW will be
      Wt  x h
Vol
                    6l x 0.7  x 2   =     4.2   x  2    = 1.40
6l                        6
Draft in water of density 1.010 = 1.40 x   1/1.010       
 = 1.386