Lesson 2: Calculating Changes in the Centre of Gravity

In Lesson 1 we learned about calculating moments and the condition for equilibrium.

When loading, discharging, or shifting cargo on a ship, the 'centre of gravity,' or condition for equilibrium, changes. As cargo is moved, the locations of masses aboard the ship change and the distance of those masses from the keel also change. All of these changes effect the moments acting on a vessel.

In Lesson 2, we will learn how to calculate the new centre of gravity for a vessel, when her cargo is loaded, discharged, or shifted.

Centre of Gravity

The centre of gravity of a body is the point at which the entire weight of the body may be considered as concentrated such that, if supported at that point, the body would remain in equilibrium. A simple example of the centre of gravity is the middle of a seesaw. Knowing where the centre of gravity (G) of a ship is located is very important because of its effect on the ship's stability.

The centre of gravity is measured from the keel of the ship. You will often be required to find the new position of 'G' when cargo is moved. The distance from the centre of gravity (G) to the keel (K) is denoted by KG. If G₁ represents the new centre of gravity then GG₁ denotes the shift in position of the centre of gravity and KG₁ represents the distance between the new centre of gravity and the keel. Also if 'g' represents the centre of gravity of cargo loaded, discharged or shifted, then 'Kg' represents the distance from the cargo to the keel.

Before moving on make sure you have done the readings for this lesson.

The following formulae are used to calcuate the shift (GG₁) and new position (KG₁) of the centre of gravity due to:

Loading new cargo moves the centre of gravity towards the added cargo $$\begin{array}{l}G{G}_1=\frac{wd}{W+w}\\ \text{where W = ship's initial displacement (mass)}\\ \text{ w = weight of cargo loaded}\\ \text{ d = vertical distance between G and the centre of gravity of the cargo weight g}\end{array}$$
Discharging cargo moves the centre of gravity away from the location of the removed cargo $$\begin{array}{l}G{G}_1=\frac{wd}{W-w}\\ \text{where W = ship's initial displacement}\\ \text{ w = weight of the cargo discharged}\\ \text{ d = vertical distance between G and g}\end{array}$$
Shifting cargo moves the centre of gravity in the same direction as the cargo $$\begin{array}{l}G{G}_1=\frac{wd}{W}\\ \text{where W = ship's displacement (including the weight being shifted) }\\ \text{ w = weight shifted}\\ \text{ d = the distance through which the weight is shifted}\end{array}$$
In each case, $$K{G}_1=KG\pm G{G}_1\text{(depending on the location of the loaded, discharged or shifted cargo)}$$

Example 1: A ship displaces 12000 t and has an initial KG of 6 m. Calculate the final KG (or KG₁) if :
a) 1000 t of cargo is loaded into the lower deck hold at Kg 3 m.
b) 1200 t of cargo is loaded onto the main deck at Kg 10 m.

Solution:

Example 2: A ship displaces 12500 t and has an initial KG of 6 m. Calculate the final KG (or KG₁) if :
a) 500 t of cargo is discharged from the lower deck hold at Kg 3 m.
b) 450 t of cargo is discharged from the main deck at Kg 10 m.

Solution:

Example 3: A ship displaces 15000 t and has an initial KG of 7 m. Calculate the final KG if a weight of 15 t is moved vertically upwards through a distance 4.5 m

Solution:

If we needed to calculate the change in centre of gravity for several simultaneous cargo changes, it would be very cumbersome to recalculate KG for each cargo change one at a time. To simplify the process we use moments. Any cargo produces a clockwise moment (+) when loaded, any discharged cargo produces anti-clockwise moment (-). Any shifted cargo is considered to produce two moments: the cargo is treated as discharged from its original position, creating a (-) moment, and added to its new position, producing a (+) moment. Finally, to simplify the diagram we draw the whole system in a horizontal direction.

Example 4: A ship with mass of 35000 tonnes has its centre of gravity 16 metres above the keel. Find the new centre of gravity if the following cargo is loaded to the ship:

160 t 18 m above the keel (or Kg = 18 m)
320 t 15 m above the keel (or Kg = 15 m)
1600 t 8 m above the keel (or Kg = 8 m)
500 t 4 m above the keel (or Kg = 4 m)

Solution:

Example 5: A ship with mass of 30500 tonnes has its centre of gravity 13 metres above the keel. Find the new centre of gravity if 200 tonnes and 2000 tonnes of cargo has been loaded 11 metres and 8 metres above the keel respectively , a 600 t of cargo has been discharged from a position 6 metres above the keel and 300 t of cargo is shifted from Kg = 3 m to Kg = 12 m.

Solution: