Problem 41 - Fundamentals of Physics

Question

What fraction of the volume of an iceberg (density \(917 \mathrm{~kg} / \mathrm{m}^{3}\) ) would be visible if the iceberg floats (a) in the ocean (salt water, density \(\left.1024 \mathrm{~kg} / \mathrm{m}^{3}\right)\) and \((\mathrm{b})\) in a river (fresh water, density \(1000 \mathrm{~kg} / \mathrm{m}^{3}\) )? (When salt water freezes to form ice, the salt is excluded. So, an iceberg could provide fresh water to a community.)

Step-by-Step Solution

Answer

In ocean water, about 10.45% is visible; in river water, about 8.3% is visible.

Step 1: Understand the Principle of Buoyancy

Consider Archimedes' principle, which states that the buoyant force on an object submerged in a fluid is equal to the weight of the fluid that the object displaces. For an object floating on a fluid, the weight of the object is equal to the weight of the fluid displaced.

Step 2: Establish the Balance for Ocean Water

For the iceberg in ocean water, let the volume of the iceberg be \( V \) with a density \( 917 \mathrm{~kg} / \mathrm{m}^{3} \), and the density of ocean water be \( 1024 \mathrm{~kg} / \mathrm{m}^{3} \). The buoyant force equals the iceberg's weight: \( 917 \times V = 1024 \times V_{displaced} \).

Step 3: Solve for Visible Volume Fraction in Ocean

From the balance equation, \( V_{displaced} = \frac{917}{1024}V \). The fraction of the iceberg submerged is \( \frac{917}{1024} \). Therefore, visible fraction = \( 1 - \frac{917}{1024} \).

Step 4: Calculate for Ocean Water

Visible fraction in ocean water is \( 1 - \frac{917}{1024} = \frac{107}{1024} \approx 0.1045 \).

Step 5: Establish the Balance for River Water

For the iceberg in river water, the density of fresh water is \( 1000 \mathrm{~kg} / \mathrm{m}^{3} \). The balance equation: \( 917 \times V = 1000 \times V_{displaced} \).

Step 6: Solve for Visible Volume Fraction in River

From the balance equation, the submerged fraction is \( \frac{917}{1000} \). Thus, the visible fraction is \( 1 - \frac{917}{1000} \).

Step 7: Calculate for River Water

The visible fraction in river water is \( 1 - \frac{917}{1000} = \frac{83}{1000} = 0.083 \).

Key Concepts

Buoyancy

Buoyancy is a fascinating force that allows objects to float in fluids. Whether it's a ship on the ocean or an iceberg in the sea, buoyancy keeps them afloat. This force arises because of Archimedes' principle, which indicates that an object submerged in a fluid experiences a force equal to the weight of the fluid it displaces.

For an object that floats, the weight of the object is balanced by the buoyant force, which is why only a portion of the object typically stays above the fluid's surface. In the case of an iceberg, understanding buoyancy helps determine how much of the iceberg remains visible above the water.

• Buoyant force = weight of the fluid displaced
• When the object's weight matches the buoyant force, it floats
• The submerged portion depends on the densities involved

Density

Density is a crucial factor in understanding buoyancy and the behavior of objects in fluids. It is defined as mass per unit volume, often represented in kilograms per cubic meter (kg/m³). The concept of density helps explain why some objects float while others sink.

In the case of the iceberg exercise, two densities are important: the density of the iceberg itself and the density of the surrounding water, whether fresh or salt.

• Density of iceberg = 917 kg/m³
• Density of saltwater = 1024 kg/m³
• Density of freshwater = 1000 kg/m³

By comparing these densities, we can calculate how much of the iceberg stays above water. A lower density iceberg floats higher above the waterline compared to the higher density water.

Volume Displacement

Volume displacement refers to the amount of fluid that is pushed aside when an object is placed in it. According to Archimedes' principle, the weight of this displaced fluid is directly related to the buoyant force experienced by the object.

When solving the problem of the iceberg, the volume of water displaced helps to calculate the visible portion of the iceberg. The volume displaced is connected to the visible and submerged parts of the iceberg by the balance of forces.

• Volume displaced is calculated using the densities
• Visible fraction = Total volume - Submerged volume
• In ocean: Submerged fraction = \( \frac{917}{1024} \)
• In river: Submerged fraction = \( \frac{917}{1000} \)

Fresh vs Salt Water

Water's properties can change depending on dissolved substances, like salt. Freshwater and saltwater have different densities, which affect how objects float.

Saltwater generally has a higher density than freshwater due to the dissolved salts. This results in larger buoyant forces in saltwater compared to freshwater. Hence, an iceberg will float slightly differently in these two types of water.

• Saltwater's higher density means more of an iceberg can remain above the surface
• With freshwater, the iceberg displaces a greater volume of water
• This difference affects the visible fraction of icebergs

Therefore, understanding these differences helps in determining how much of an iceberg, or any floating object, is visible in different types of bodies of water.