Calculating the Buoyancy of the Titanic
Will the Titanic actually sink with five compartments flooded, and can she sink with less or more compartments flooded? Archimedes’ principle can help us determine the answer. Remember that, for simplicity, we are assuming that the Titanic has a displacement of 50 000 metric tonnes.
There are two external forces acting on the Titanic (assuming that she is stationary in the water). The weight (the force of gravity acting on the mass of the ship) acts downwards, and the buoyant force (due to the displacement of the water) acts upwards. When the ship is floating and has no acceleration in the vertical direction, then the ship is in equilibrium, and the two forces exactly balance out. When the ship takes in water, the buoyant force is less than the weight. There is a net force in the downward direction that is acting on the ship, and the ship will sink lower in the water.
Shown below is a representation of the box-shaped Titanic floating in the water. In its current state, the ship will be displacing water to provide a buoyant force that is exactly equal in magnitude and opposite in direction to the force of gravity. The volume of water displaced by the ship is equal to the volume of the keel that is beneath the waterline.
It is helpful at this point to draw a simplified diagram of the forces acting on the Titanic. This is known as a free body diagram. In a free body diagram, we reduce the Titanic to a single point and draw the forces acting on it.
When not flooded and floating normally, the Titanic displaces an amount of water weighing approximately 50 000 t, or 50 000 m3 of water. Of course, the Titanic could displace more than 50 000 t of water and still float. We need to find the maximum amount of water that she can take in before she sinks.
In reality, the Titanic would tilt downwards on the side where the water flooded, as shown above. Our calculations will neglect this behaviour. We will instead assume that the Titanic floods uniformly and will not tilt when sinking, as shown below.
As stated earlier, the Titanic’s watertight compartments were not totally waterproof. All of the compartments could eventually fill up, if the ship sank deep enough into the water so that the waterline exceeded the height of the bulkheads. The bulkheads span several decks, and we will need to assume that the floor and room partitions don’t hold back or sequester water at all.
The Titanic’s first five compartments (closest to the bow) were opened by the iceberg. (In reality, six compartments were ruptured, but the sixth had a such a small scratch that the water flooding it could be extracted by the Titanic’s pumps.) Each compartment has a volume of 50 000 m3 ÷ 16 = 3125 m3, so each compartment could hold 3125 t of water. The five compartments ruptured by the iceberg, when filled, would hold 3125 × 5 = 15625 t of water.
Now the Titanic will be displacing 65625 t of water, a volume of 65 625 m3. We can use this to determine the new draught of the Titanic. We know that (for a box) Volume = area of bottom × draught of ship So we see that the area of the bottom of the (box-shaped) Titanic is 50 000 m3 ÷ 10.5 m = 4760 m2. The draught of the Titanic with five flooded compartments is 65 625 m3 ÷ 4 760 m2 = 13.8 m. Water at this level will easily spill over the bulkheads and flood the entire ship.
It is important to note that these calculations would be correct only if the ship had flooded uniformly. In reality, the ship would have tilted, and it would not have needed to be filled with as large a weight in water before the waterline exceeded the height of the bulkheads.