Why Is Most of an Iceberg Under Water? And How Much?

Original uncaptioned image Lurens, public domain image via Pixabay.com

Ninety Percent of an Iceberg Is Below the Waterline

Icebergs can be massive, but did you know that you're only seeing literally the tip of them above the water? Yes, up to 90% of an iceberg's volume can be under water. But why does that amount sink and not just float above sea level like the rest of it? In this article we explore why.

The Archimedean Principle

A body wholly or partially immersed in a fluid experiences an upthrust or buoyant force, equal to the weight of the displaced fluid.

In a previous science tutorial called Archimedes' Principle, Buoyancy Experiments and Flotation Force, we discovered how objects placed in a liquid experience a force called buoyancy.

The Archimedean Principle or Principle of Archimedes simply states that when you put something into a fluid (A fluid can be a liquid like water or gas like air), the force acting upwards on the object equals the weight of fluid that was displaced. Imagine a container full to the brim with water. If something is put into the water in the container, it sinks a bit and floats, totally submerges and floats, or completely sinks to the bottom. The amount of water that flows over the brim is what is displaced. If that water is weighed, it equals the buoyant force pushing up on the object in the water.

The principle of Archimedes. Buoyant force equals the weight of the displaced liquid. The weight measured by the scales is 6kg less 2 kg = 4 kg. © Eugene Brennan

Working Out the Volume of an Iceberg That's Under the Water

In the image below, we have an iceberg with a lot of its volume under the water. To do the calculations, first let's assign some variables so that we can work out the equations.

V is the total volume of the iceberg

ρ_i is the density of fresh water ice

ρ_s is the density of seawater

k is the fraction of the berg that is underwater

g is the acceleration due to gravity

m is the mass of the displaced water

W_d is the weight of displaced water

W_i is the weight of the iceberg

Calculations

The total volume of the iceberg is V.

The variable k represents the proportion of the berg that is below the water. k has to be greater than or equal to 0 and less than or equal to 1. In other words, between none and all of the berg lies below the water line. We don't know what k is and it's this variable that we need to work out an equation for.

Calculation of mass of displaced water

The volume of the berg below water = kV
and since the ice displaces its own volume, the volume of displaced water is also kV

The Archimedes Principle tells us that the upthrust or buoyancy force = weight of the displaced water. We need to work out the weight of this water.

We defined a variable ρ_s as the density of sea water

The volume of the displaced water = the volume of the berg below water = kV

We know that density = mass / volume, so mass = density x volume

So mass of the displaced water m = density of sea water x volume of water displaced.

m = ρ_skV .......................Eqn (1)

Calculation of weight of displaced water

Now since Archimedes Principle specifies weight, we convert mass to weight by multiplying by the acceleration due to gravity

So weight = mg and substituting for m from Eqn (1), the weight of displaced water

W_d = ρ_skVg .....................Eqn (2)

Calculation of weight of iceberg

Ice sheets are made from snow that gets packed over long periods of time and converted into ice. Bits of ice eventually break off from the sheets and form floating icebergs. Since icebergs came from snow that originated as water that evaporated off land and sea, they're made of fresh water, not sea water. Pure fresh water has a density of 1 g/cm³. Ice is less dense than water however, because when water freezes, its volume increases. Since density = mass / volume and mass stays the same, but volume increases, density decreases. The density of ice ranges from 0.9167 to 0.9168 g/cm3

Assuming the berg is made of fresh water and its volume is V and the density of fresh water ice is ρ_i, then the weight W_i of the berg is

W_i = ρ_iVg .............................Eqn (3)

© Eugene Brennan

Working out k, the proportion of the iceberg under water

Now put Eqn (2) and Eqn (3) together. Since the berg is floating, its weight acting downwards (A in the diagram above) is balanced by the upthrust (force B), equalling the weight of the displaced water)

So upthrust = W_d = weight of iceberg = W_i

Substituting for the values of W_d and W_i from Eqn (2) and Eqn (3) gives:

ρ_skVg = ρ_iVg

Simplifying by dividing both sides of the equation by Vg gives us

ρ_sk = ρ_i

or k = ρ_i / ρ_s

So now two things become apparent:

• The proportion k of the berg below water is independent of the size of the berg.
• k doesn't depend on the shape of the iceberg.

The proportion of an iceberg below sea level is simply the ratio of the density of fresh water ice to the density of sea water. If the density of ice is 0.9167 g/cm³ and the density of sea water is on average 1.025 g/cm³ . then

k = 0.9167 / 1.025 = 0.89.

So approximately nine tenths or 90% of an iceberg is under water.

How Much of an Iceberg Is Under Water?

Approximately 90% of an iceberg is under water

Why Is Most of an Iceberg Below Water?

We worked out the numbers mathematically above, but intuitively what is the reason? It's simply because the density of freshwater ice is so close to the density of seawater. So a lot of the iceberg has to sink below the surface to create enough displacement to generate the buoyancy force to balance the weight of the berg to keep it afloat.

If the density of ice equalled that of seawater, the berg would have to submerge completely to generate a buoyancy force to keep itself afloat. If the density was higher again (e.g., if it had lots of rock or gravel embedded inside), the weight of the displaced water would be less than the weight of the berg, even if fully submerged, and it would sink. If an object is made of styrofoam (expanded polystyrene), it only has to displace a little water in proportion to its volume to keep itself afloat and most of its bulk will be above water.

What is a Hydrometer?

A hydrometer is a measuring instrument used to measure the density of a liquid. By measuring density, information can be ascertained about the state of a process. For instance, hydrometers are sometimes used to check the concentration of sulphuric acid in a lead acid battery under charge. If the concentration isn't up to a certain level after several hours, it's an indication that the battery has aged and is no longer performing well. Hydrometers are also used in home brewing and wine making to measure alcohol content of the mix.

How Does a Hydrometer Work?

We established that the proportion of an iceberg under water is simply the ratio of the density of the ice in the iceberg to the density of seawater. The formula for proportion applies to all scenarios when an object is placed in another medium, be it liquid or gas. The formula is for volume, but by making a hydrometer cylindrical, the depth the hydrometer sinks to can be made to represent the density of the liquid. A scale on the side of the hydrometer allows density to be read directly.

Disclaimer 

This content is accurate and true to the best of the author’s knowledge and is not meant to substitute for formal and individualized advice from a qualified professional.

© 2022 Eugene Brennan

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