Understanding the term “displacement”
Under normal conditions, water is an incompressible fluid, meaning that its density and volume remain constant regardless of pressure or applied forces. When a finite object is placed in water, two things happen:
The object exerts a force on the water medium due to its weight.
The object tends to displace a certain volume of water with which it is in direct contact.
These two phenomena are the basis of all our explanations.
When the first happens, the fluid or water medium tends to produce a reaction force equal to the weight of the object, thus producing a downward force, according to Newton’s third law of motion (for every action there is an equal and opposite reaction).
However, just like solids, if the first point were the only law for fluids, then a small coin or a screw weighing only a few grams would be easily balanced by a large amount of water. But due to the properties of fluids, the first point depends on the second point.
What does the second point mean technically? When an object is placed in water, it occupies a certain volume in the fluid domain. Since the total volume of the system must remain constant according to first principles, an equivalent, uniform overall increase in the volume of the surrounding water offsets this loss or local “displacement” of volume or displaced space.
Now, note that we use vague terms like “a certain volume” or “a certain volume” when discussing displacement in water. How do we ultimately define this displacement volume? The answer is that it depends on several factors.
This is because, when a given object is placed in water or any other fluid, its displacement volume depends on factors such as:
- The weight of the object
- The geometry and physical shape of the object
- The physical properties of the fluid (density)
While we discussed the mass of an object, represented by weight in the physical world, in the world of fluids, the geometry and shape of the object are more important.
Imagine placing a wooden box, weighing 5 kg and measuring 2m x 2m x 2m, into a small tank of fresh water (density = 1 kg/m³). When a 5 kg object (equivalent to 5 times the acceleration due to gravity, g) is placed in water, it will begin to sink and then float up slowly.
This is because it has reached a state of mechanical equilibrium, and the reaction force of the fluid medium is exactly equal to the object’s weight. Since water is incompressible and extremely elastic, how is this reaction force generated? Or more accurately, where does it come from?
The source of the reaction force is the volume of the space occupied by the wooden box, or in other words, the volume of the fluid displaced multiplied by the density (mass) multiplied by the acceleration of gravity.
What is the volume of this space? Length × width × depth. Since water is an ideal fluid, it can take any shape depending on the object with which it interacts. Therefore, when the box is placed, the length and width of the volume of the water ball displaced are also equal, that is, 2 × 2. How deep is the water ball? This is essentially the depth to which the box sinks, which is the key to our discussion (the variable “t”). The volume of water displaced, or conversely, the space occupied by the box in the fluid field, is equal to: 2 x 2 x t.
To achieve mechanical equilibrium, and to account for the equivalence of forces, the force of gravity acting on the box (i.e., 5 times the force of gravity) must equal the reaction force of the water, i.e., volume displaced x density x gravity.
By solving the equilibrium, the box sinks to a depth of 1.25 meters. This is referred to as the immersion depth of the object. Therefore, the net volume occupied by the box in the fluid medium, or conversely, the volume of water displaced by the fluid, is 2 x 2 x 1.25 = 5 cubic meters.
Since the density is 1, the reaction force of the water is 5 times the force of gravity exerted by the box. Due to this equilibrium, the object floats in the water.
This is the famous Archimedean principle. An object floating in water, whether partially or completely, is subject to an upward force called the buoyancy force. Mathematically, this force is equal to the weight of the water displaced by the object plus the weight of the object itself.
From another perspective, the column or volume of water that displaces the object and is evenly distributed elsewhere weighs the same as the object. Therefore, the water field exerts an equal and opposite force that keeps the system in equilibrium.
This column or volume of water is referred to as displacement, or more precisely, volumetric displacement. When discussing mass, the weight of the water displaced is equal to the physical weight of the object (quantified in terms of mass) and is referred to as mass displacement.
In the previous example, the mass displacement of the box is 5 kg, and its volume is five cubic meters.
We should place a 4 kg metal rod in the box. What happens? The mass of the entire system becomes 9 kg. According to the previous formula, the value of the unknown variable t is 2.25 meters.
However, this exceeds the maximum vertical extension of the box, which is 2 meters. This means that the water level is above the top of the box, but an additional 0.25 meters is required to reach equilibrium.
The water immediately enters the box, and the entire system immediately sinks. Therefore, by adding 4 kg mass, the design of this system does not comply with mechanical equilibrium or the law of buoyancy.
Now, we have another heavier box, weighing 8 kg, made of the same material and design, but with dimensions of 3 x 3 x 2 and weighing 4 kg.
According to the above formula, (8 + 4) × g = (3 x 3 x t) × 1 × g.
The draft is about 1.33 meters, which means it floats comfortably with about 0.67 meters to spare.
There are two main results of this one-meter increase in size:
Better mass distribution, or density, means the system has less mass per unit volume.
The volume displaced underwater, or displacement, is the amount of weight (and buoyancy) that is sufficient to balance the weight of the object.
Therefore, the mass of the object is not taken into account when calculating buoyancy, and equilibrium is only achieved when the shape and design of the object contribute to the balance of forces. This is why metal ships weighing thousands of tons float, while small propellers or metal poles will sink immediately.
Ship displacement
When we say that a ship has a displacement of 1,000 tons, what we mean is simple:
Its actual weight on land is 1,000 tons.
When the ship floats, this weight appears as displacement in the water. Therefore, the displacement is equivalent to 1,000 tons and creates an equal and opposite effect, which is called buoyancy.
Lightweight and Deadweight
The displacement of a ship consists of two parts: lightweight and deadweight.
The lightweight of a ship is its empty weight. This includes the weight of the hull structure, machinery, fittings, and all permanent and non-consumable parts.
The deadweight of a ship is the sum of the weight of all consumables, including cargo, fuel, ballast water, crew and passengers, stores, and any other consumables. This is the deadweight of the ship.
When a ship reaches its maximum capacity, it is considered fully loaded. In this state, the ship tends to sink deeper due to its greater weight, displacing more underwater volume that provides the buoyancy required to maintain equilibrium.
Therefore, the greater the displacement, the greater the draft of the ship. This condition is referred to as the fully loaded or high seas state.
A lightly loaded ship is a ship with a minimal amount of cargo (excluding cargo and most other consumables). Due to its light weight, even a smaller volume of water (equal to the weight of the ship) can provide the necessary buoyancy (equal to the weight) to keep the ship upright.
Density
Density also plays a very important role in determining the buoyancy mechanism of a ship. It is worth noting that in the previous formula, the density is one because it is freshwater. Now, if we consider seawater with a density of 1.025 kg/m³, the formula becomes:
5 × g = (2 × 2 × T) × 1.025 × g
At this point, the T value is approximately 1.21, which is less than 1.25 in freshwater. This actually means that the same object sinks less quickly in salt water than in fresh water.
Continuing with the above idea, due to density, this means that for a floating object of the same weight, a larger amount of water needs to be displaced in a freshwater environment than in a saltwater environment to maintain the necessary balance. Therefore, the higher the density, the smaller the displacement of the ship, and vice versa.