Problem 34 - Fundamentals of Physics

Question

A boat floating in fresh water displaces water weighing \(35.6 \mathrm{kN}\). (a) What is the weight of the water this boat displaces when floating in salt water of density \(1.10 \times 10^{3} \mathrm{~kg} / \mathrm{m}^{3}\) ? (b) What is the difference between the volume of fresh water displaced and the volume of salt water displaced?

Step-by-Step Solution

Answer

(a) 35.6 kN, (b) 0.33 m³ difference.

Step 1: Understand the Problem

The weight of the water displaced by the boat in fresh water is given as \(35.6 \text{kN}\). We are asked to find the weight of the water displaced when the boat is in salt water and the volume difference between the fresh water and salt water displaced.

Step 2: Weight of Salt Water Displaced

According to Archimedes' principle, the weight of the water displaced by the boat is equal to the weight of the boat. This means that the weight of the salt water displaced will also be \(35.6 \text{kN}\). The weight of the displaced water depends on the weight of the boat and is thus unchanged from fresh water to salt water.

Step 3: Volume of Fresh Water Displaced

To find the volume of fresh water displaced, use the formula:\[ \text{Weight} = \text{Volume} \times \text{Density} \times g \]where the density of fresh water is \(1000 \text{kg/m}^3\) and \(g = 9.8 \text{m/s}^2\). Thus,\[ 35.6 \times 10^3 = V_{fw} \times 1000 \times 9.8 \]Solve for \(V_{fw}\):\[ V_{fw} = \frac{35.6 \times 10^3}{1000 \times 9.8} \approx 3.63 \text{m}^3 \]

Step 4: Volume of Salt Water Displaced

For the salt water of density \(1.10 \times 10^{3} \text{kg/m}^3\):\[ 35.6 \times 10^3 = V_{sw} \times 1.10 \times 10^3 \times 9.8 \]Solve for \(V_{sw}\):\[ V_{sw} = \frac{35.6 \times 10^3}{1.10 \times 10^3 \times 9.8} \approx 3.30 \text{m}^3 \]

Step 5: Calculate the Difference in Volume

Calculate the difference in the volume of fresh water and salt water displaced:\[ \text{Volume Difference} = V_{fw} - V_{sw} = 3.63 \text{m}^3 - 3.30 \text{m}^3 = 0.33 \text{m}^3 \]

Key Concepts

Density of Water

Density is a measure of how much mass is contained in a given volume. For liquids, it's often expressed in units like kilograms per cubic meter (kg/m³). Fresh water has a density of about \(1000 \text{ kg/m}^3\), while salt water is a bit denser due to the dissolved salts, with a density of around \(1.10 \times 10^3 \text{ kg/m}^3\).

Density plays a key role in many aspects of physics and engineering, especially when it involves liquids and buoyancy. When an object, such as a boat, is placed in water, the density of the water affects the volume of water the boat displaces. This displaced water needs to equal the weight of the boat for it to float. In our exercise, the boat displaces a certain volume of fresh water, and when it moves to salt water, the volume changes because the density is different.

The relationship between density and volume can be understood through the formula for weight:

• Weight = Volume × Density × g (gravitational acceleration)

For the boat in fresh water, we calculated this relationship, and similarly, for salt water, we used the same principle. Understanding the density helps in calculating how much of the water is displaced and why different densities will displace different volumes.

Buoyancy

Buoyancy is the force that makes objects float in a fluid. It's the upward force exerted by the fluid on any object placed in it. This force is a result of Archimedes' principle, which states that the buoyant force is equal to the weight of the fluid displaced by the object.

In practice, for a floating object like a boat, the weight of the displaced water gives us the buoyant force counteracting the weight of the boat. If the weight of the boat is greater than the buoyant force, the boat will sink. If they are equal, the boat will float.

When a boat moves from fresh water to salt water, the buoyant force has to balance with the weight of the boat, which stays constant. Therefore, the boat, although heavy, displaces less volume of salt water due to its higher density. This helps illustrate the concept of buoyancy further:

• Buoyant force = Weight of displaced water
• Determines floating or sinking behavior
• Affected by density changes in the fluid

This tells us why the same boat floats differently in salty water compared to freshwater.

Fluid Mechanics

Fluid mechanics is a branch of physics that studies the behavior of fluids (liquids and gases) and the forces acting on them. Understanding fluid mechanics is crucial for explaining why boats float, airplanes fly, and blood flows through the human body. It encompasses concepts like pressure, flow, and of course, buoyancy and density.

In the context of our exercise, fluid mechanics explains how the densities of different fluids affect the buoyant force and the volume of fluid displaced by floating objects.

• Pressure in fluids increases with depth: It's why deep-diving submarines need to be strong.
• Flow deals with how fluids move: essential for understanding things like river currents or air over a wing.

When looking at a boat and its displacement of water, fluid mechanics provides the tools to calculate the impact of different fluid properties, like the density differences between fresh and salt water, and consequently, how they affect buoyancy performance. Fluid mechanics is a vast field, but its principles help explain many everyday phenomena and also industrial applications, allowing engineers to design better ships, airplanes, and even improve water distribution systems.