Question

Ice has a density of roughly \(0.9 \mathrm{~g} / \mathrm{cm}^{3}\) and water has a density of about \(1.0 \mathrm{~g} / \mathrm{cm}^{3}\). Use that information to complete the following: a. Using the principle of isostasy, approximate how much of a 100 -meter-thick iceberg would be located above sea level. b. Assuming that this iceberg melts evenly, how much of it will be above sea level after half of its mass has melted? c. Do you think icebergs make good examples for the principle of isostasy? Explain. d. Compare the melting of an iceberg to the process of isostatic adjustment that occurs when a mountain erodes.

Short answer

a. 10 meters above water; b. 5 meters above water; c. Yes, icebergs illustrate isostasy well; d. Melting icebergs and eroding mountains both adjust isostatically.

Step-by-step solution

Determine Buoyancy

According to Archimedes' principle, an object floats when the weight of the fluid displaced equals the weight of the object. If ice has a density of 0.9 g/cm³ and water 1.0 g/cm³, then 90% of the ice is submerged when floating. This leaves 10% above water.

Calculate Above-Water Portion of Iceberg

If the iceberg is 100 meters thick, and 90% is submerged, calculate the part above water: \[0.1 \times 100 = 10 \text{ meters}\] Thus, 10 meters of the iceberg is above sea level.

Calculate Remaining Iceberg Volume After Melting

After half the mass is melted, the iceberg retains half its original mass. Since density remains unchanged, the iceberg's volume is halved. Originally, 90% was submerged, so half of the remaining iceberg's reduced volume stays submerged.

Evaluate Water Line Shift After Melting

With half the volume and weight reduction, \[0.1 \times \frac{100}{2} = 5 \text{ meters}\]Thus, 5 meters of iceberg remains above water after half its mass melts.

Evaluate Isostasy Example Validity

Icebergs are good examples of isostasy since they float by displacing sea water equal to their weight, similar to how Earth's crust floats on denser mantle. Hence, they model isostatic balance well.

Compare Iceberg Melting to Isostatic Adjustment

When an iceberg melts, its loss in mass results in re-balancing to maintain buoyancy, much like isostatic adjustment. Eroding mountains reduce weight, causing the crust beneath to slowly rebound upwards, similar to how melting icebergs adjust.

Key concepts

Buoyancy

The concept of buoyancy describes why objects float or sink in a fluid. It is based on the pressure exerted by the fluid on an object submerged in it. The key principle behind buoyancy is that the upward force or buoyant force acting on an object submerged in a fluid is equal to the weight of the fluid displaced by that object.
This means that if an object displaces more fluid than it weighs itself, it will float. Conversely, if it weighs more than the displaced fluid, it will sink.
For an iceberg floating in the ocean, it displaces an amount of water equal to its own weight, keeping it afloat. Because the density of ice is less than that of water, only a portion of the iceberg needs to displace enough water to equal its weight. This results in some of the iceberg being submerged while the rest remains above water.

Archimedes' Principle

Archimedes' Principle is a fundamental law of physics explaining how buoyancy works. It states that any object, whether wholly or partially immersed in a fluid, is buoyed up by a force equal to the weight of the fluid it displaces.
When applied to icebergs, this principle explains why they float. Considering an iceberg with a density of 0.9 g/cm³, compared to water's density of 1.0 g/cm³, Archimedes’ principle allows us to deduce that 90% of the iceberg's volume must be submerged to displace water weighing as much as the iceberg itself.
The remaining 10% of the iceberg's volume will be above the water's surface. This illustrates how Archimedes' Principle provides a clear explanation of the balance between the submerged and exposed portions of floating objects.

Density of Ice and Water

The densities of ice and water are crucial in understanding why icebergs float and how much of them remain above water. Density is a measure of mass per unit volume. For ice, this is about 0.9 g/cm³, while for water it’s around 1.0 g/cm³.
Due to ice being less dense than water, it floats. The 0.9 g/cm³ density implies that ice is 90% as dense as water, leading to 90% of an iceberg being submerged when it floats.
This difference in density also explains why only 10% of a floating iceberg appears above the surface, providing a simple yet profound insight into the natural design of our world. This concept is pivotal for calculations involving floating bodies and their stability.

Isostatic Adjustment

Isostatic adjustment is the process by which the Earth's crust maintains a balanced state, akin to floating on the denser mantle. This geological phenomenon closely relates to how an iceberg behaves in water.
When an iceberg melts, it loses mass and volume, causing it to re-adjust vertically in water to maintain buoyancy—this is similar to how isostatic adjustment operates.
When, for example, a mountain erodes, the reduction in mass causes the surface to rise gradually, which can be compared to how the remaining mass of a melting iceberg allows it to stay afloat with a different above-water profile.

• This is an example of isostatic balance where changes on the surface trigger adjustments to maintain equilibrium
• Icebergs exemplify this concept as they melt and re-balance, similar to crustal adjustments in response to changing surface loads

These interactions highlight the fundamental principles of isostasy, tying back to both natural ice phenomena and long-term geological processes.