The center of gravity can be imagined to be a point through which all of the vessel's weight acts vertically downward.

The addition of weight causes the center of gravity to move in various directions. Therefore the position of the center of gravity is dependent upon the size of the weights added to the vessel, and the position in which they are added. In other words, the final position of the center of gravity is dependent upon the practices of the vessel operator.

The following three rules describe the movement of the center of gravity of the vessel.

(1) The center of gravity moves toward an added weight. (See Fig A)

Figure A - Weight Added

(2) The center of gravity moves away from a discharged weight. (See Fig B)

Figure B - Weight Removed

(3) The center of gravity moves parallel to the movement of a weight which is already on board. (See Fig C)

Figure C - Shifting Weights

The magnitude of the movement of the center of gravity is directly dependent upon:

(1) The mass of the weight involved; (2) The distance between the center of gravity of the vessel and the center of gravity of the weight. (3) The displacement of the vessel.

Shift in center of gravity due to addition of mass

When a mass is added to a ship, the center of gravity of the ship moves toward the added mass. The distance moved by the ship's center of gravity depends upon the magnitude of the added mass, the distance of the mass from the ship's center of gravity, and the displacement of the ship. e.g., If a mass is placed on the port side of the ship in the forecastle, the center of gravity moves forward, upward, and to port. The actual distance and direction of this movement are seldom required, but the separate components are most important, i.e. the longitudinal, vertical and transverse distances moved. When an item on a ship is removed, the center of gravity moves away from the original position of that item.

Example: A ship of 4000 tonnes displacement has its center of gravity 1.5 m aft of midships and 4 m above the keel. 200 tonnes of cargo are now added 45 m forward of midships and 12 m above the keel. Calculate the new position of the center of gravity.

Taking moments about the midship axis (xx’): Moment aft of midships = 4000 t × 1.5 m = 6000 tm Moment fwd of midships = 200 t × 45 m = 9000 tm Excess moment forward = 9000 - 6000 = 3000 tm

Center of gravity from midships = (Excess moment) / (Total displacement) = 3000 tm / (4000 + 200) t = 0.714 m forward from the midship axis

Taking moments about the keel: Center of gravity from keel (KG) = ((4000 t × 4 m) + (200 t × 12 m)) / (4000 + 200) t = (16000 + 2400) / 4200 KG = 4.381 m

Thus the center of gravity rises (4.381 - 4.0) = 0.381 m due to the added weight.

The same answer may be obtained by taking moments about the original center of gravity, thus: Moment of ship about Center of gravity (yy’) = 4000 t × 0 = 0 tm Moment of added mass about Center of gravity = 200 t × (12 - 4) m = 1600 tm Rise in Center of gravity = (Total moment) / (Total displacement) = 1600 tm / 4200 t = 0.381 m

If the actual distance moved by the center of gravity is required, it may be found from the longitudinal and vertical movements. Longitudinal shift in the center of gravity (GT) = (1.5 + 0.714) m = 2.214 m

GG₁ = √(GT² + TG₁²) = √(2.214² + 0.381²) = 2.247 m

The angle (θ) which the center of gravity moves relative to the horizontal may be found from the above fig. tan θ = 0.381 / 2.214 = 0.1721 From which θ = 9° 46'

Shift in center of gravity due to movement of mass

When a mass which is already on board a ship is moved in any direction, there is a corresponding movement in the center of gravity of the ship in the same direction.

Consider a system composed of masses of m₁, m₂, and m₃ as shown in Fig A above, the center of gravity of each being h₁, h₂, and h₃ respectively from the base 0-0’. The distance of the center of gravity of the system from the base may be determined by dividing the total moment of mass about 0-0’ by the total mass.

Center of gravity from 0-0’ = (Total moment of mass) / (Total mass) = (m₁h₁ + m₂h₂ + m₃h₃) / (m₁ + m₂ + m₃) = y meters

If m₃ is now raised through a distance 'd' to the position shown in above Fig B, the center of gravity of the system also will rise. New center of gravity from 0-0’ = (m₁h₁ + m₂h₂ + m₃(h₃ + d)) / (m₁ + m₂ + m₃) = (m₁h₁ + m₂h₂ + m₃h₃ + m₃d) / (m₁ + m₂ + m₃) = [(m₁h₁ + m₂h₂ + m₃h₃) / (m₁ + m₂ + m₃)] + [m₃d / (m₁ + m₂ + m₃)] = y + [m₃d / (m₁ + m₂ + m₃)]

Thus it may be seen that:

Distance shift in center of gravity = m₃d / (m₁ + m₂ + m₃) Or, Shift in center of gravity = (mass moved × distance moved) / (TOTAL mass)

The center of gravity of the ship moves in the same direction as the center of gravity of the mass. Thus if a mass is moved forward and down, the center of gravity of the ship also moves forward and down.

Example. A ship of 5000 tonnes displacement has a mass of 200 tonnes on the fore deck 55 m forward of midships. Calculate the shift in the center of gravity of the ship if the mass is moved to a position 8 m forward of midships.

Shift in center of gravity = (mass moved × distance moved) / Displacement = (200 t × (55 - 8) m) / 5000 t = (200 × 47) / 5000 = 1.88 m aft

Tutorials

1. The center of gravity of a ship of 5000 tonnes displacement is 6 m above the keel and 1.5 m forward of midships. Calculate the new position of the center of gravity if 500 tonnes of cargo are placed in the ‘tween decks 10 m above the keel and 36 m aft of midships.

2. A ship has 300 tonnes of cargo in the hold, 24 m forward of midships. The displacement of the vessel is 6000 tonnes and its center of gravity is 1.2 m forward of midships. Find the new position of the center of gravity if this cargo is moved to an after hold, 40 m from midships.

3. An oil tanker of 17000 tonnes displacement has its center of gravity 1 m aft of midships and has 250 tonnes of oil fuel in its forward deep tank 75 m from midships. This fuel is transferred to the after oil fuel bunker whose center is 50 m from midships. 200 tonnes of fuel from the after bunker is now burned. Calculate the new position of the center of gravity: (a) After the oil has been transferred (b) After the oil has been used.

4. A ship of 3000 tonnes displacement has 500 tonnes of cargo on board. This cargo is lowered 3 m and an additional 500 tonnes of cargo is taken on board 3 m vertically above the original position of the center of gravity. Determine the alteration in position of the center of gravity.