Question

The density (mass per unit volume) of ice is \(917 \mathrm{~kg} / \mathrm{m}^{3}\) and the density of seawater is \(1024 \mathrm{~kg} / \mathrm{m}^{3}\). Only \(10.4 \%\) of the volume of an iceberg is above the water's surface. If the volume of a particular iceberg that is above water is \(4205.3 \mathrm{~m}^{3},\) what is the magnitude of the force that the seawater exerts on this iceberg?

Short answer

Question: Calculate the magnitude of the force that seawater exerts on an iceberg with 10.4% of its volume above water and an above-water volume of 4205.3 m³. The density of ice is 917 kg/m³, and the density of seawater is 1024 kg/m³. Answer: The magnitude of the force that the seawater exerts on the iceberg is approximately \(363,293,758.27~\mathrm{N}\).

Step-by-step solution

Calculate the volume of the entire iceberg

As 10.4% of the iceberg's volume is above the water's surface, and the above-water volume is 4205.3 m³, we need to find the total volume of the iceberg. Let the total volume be V. We can set up the equation: \(0.104 \times V = 4205.3~\mathrm{m^3}\). To find V, divide both sides of the equation by 0.104. \( V = \frac{4205.3}{0.104} \approx 40437.98~\mathrm{m^3}\).

Calculate the mass of the iceberg and displaced water

Using the density of ice (917 kg/m³), multiply it by the total volume of the iceberg to find its mass: \(m_\text{iceberg} = 917 \times 40437.98 = 37060186.14 \mathrm{~kg}\). As the volume of the iceberg submerged under water displaces an equal volume of seawater, we know the volume of displaced seawater is \(40437.98 - 4205.3 = 36232.68~\mathrm{m^3}\). Multiply this volume by the density of seawater (1024 kg/m³) to find the mass of the displaced seawater: \(m_\text{seawater} = 1024 \times 36232.68 = 37062223.04 \mathrm{~kg}\).

Calculate the force exerted by seawater on the iceberg (buoyant force)

According to Archimedes' principle, the buoyant force on an object submerged in a fluid is equal to the weight of the fluid displaced by the object. The force can be calculated using the equation: \(F_\text{buoyant} = m_\text{seawater} \times g\), where g is the acceleration due to gravity (\(9.81~\mathrm{m/s^2}\)). Calculate the force: \(F_\text{buoyant} = 37062223.04 \times 9.81 \approx 363293758.27~\mathrm{N}\). The magnitude of the force that the seawater exerts on the iceberg is approximately \(363,293,758.27~\mathrm{N}\).

Key concepts

Archimedes' Principle

Understanding the Archimedes' principle is essential for solving problems related to buoyancy and flotation. Simply put, this principle states that any object, partially or completely submerged in a fluid, is buoyed up by a force equal to the weight of the fluid that is displaced by the object.

This principle helps us comprehend why objects float or sink. It's the reason why a heavy ship made of steel can float on water, while a small pebble sinks to the bottom. The ship displaces a large volume of water, creating a buoyant force greater than its weight, whereas the pebble, small in size, displaces little water and sinks as the buoyant force is less than its weight.

Real-World Application

Using Archimedes' principle, we can determine the buoyant force acting on an iceberg, as seen in our textbook example. It's the same force that allows hot air balloons to rise and submarines to adjust their buoyancy to ascend or descend in water.

Density

Density is a measure of how much mass is contained in a given unit volume of a substance or object (mass per unit volume). It is commonly expressed in kilograms per cubic meter (kg/m³) in the metric system. Different substances have different densities, and understanding this concept is vital for analyzing situations involving sinking and floating.

Density can be thought of as the 'compactness' of a material. Objects with higher density have more mass packed into the same volume as objects with lower density. In our exercise, the densities of ice and seawater are crucial to calculate the mass of the iceberg and the water it displaces, which in turn is used to determine the buoyant force.

Importance in Buoyancy

Knowing the density of two interacting substances in a buoyancy scenario allows us to predict which will float and which will sink. The iceberg, being less dense than the seawater, floats, with only a portion of its volume submerged.

Mass and Volume Relationship

The relationship between mass and volume is a foundational concept in physics and chemistry. Mass is the measure of the amount of matter in an object, while volume measures how much space that object occupies. The relationship between these two properties is what we often use to calculate density.

In many scientific problems, we use the formula \( Density = \frac{Mass}{Volume} \) to find out one of the missing quantities if the other two are known. This is particularly helpful when we need to calculate the mass of a substance from its volume and density, as we see in the solution to calculate the mass of the iceberg and displaced seawater.

Resolving the Iceberg Exercise

The mass and volume relationship is used to ascertain the total mass of the iceberg and displaced seawater. By understanding this relationship, students can grasp how the volumes of the iceberg and seawater translate into weights, which are necessary to compute the buoyant force in the exercise.