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Chapter 9: Problem 5

What is buoyancy force? Compare the relative magnitudes of the buoyancy force acting on a body immersed in these mediums: \((a)\) air, \((b)\) water, \((c)\) mercury, and \((d)\) an evacuated chamber.

Short Answer

Expert verified
Question: Arrange the following mediums in decreasing order of buoyancy force acting on an object of the same volume submerged in each medium: air, water, mercury, and an evacuated chamber. Answer: Mercury, Water, Air, Evacuated Chamber

Step by step solution

01

Understand Archimedes' Principle

Archimedes' principle states that the upward buoyancy force (F_b) exerted on an object immersed in a fluid equals the weight of the fluid displaced by the object. Mathematically, it can be expressed as: F_b = 蟻 脳 V 脳 g, where 蟻 is the fluid density, V is the volume of the fluid displaced by the object, and g is the gravitational acceleration.
02

Find the density of each medium

The density 蟻 of each medium is as follows: - Air: \(\rho_a \approx 1.225 kg/m^3\) - Water: \(\rho_w \approx 1,000 kg/m^3\) - Mercury: \(\rho_m \approx 13,600 kg/m^3\) - Evacuated chamber: In an evacuated chamber, there's no fluid, so the density 蟻 in this case is assumed to be 0 kg/m^3.
03

Compare buoyancy forces in different mediums

Now, we will compare the buoyancy forces for a body of the same volume, V, immersed in these different mediums. The buoyancy force acting on the body can be calculated using the Archimedes' principle (F_b = 蟻 脳 V 脳 g). (a) Air: \(F_{b_{air}} = \rho_a 脳 V 脳 g = 1.225 kg/m^3 脳 V 脳 g\) (b) Water: \(F_{b_{water}} = \rho_w 脳 V 脳 g = 1,000 kg/m^3 脳 V 脳 g\) (c) Mercury: \(F_{b_{mercury}} = \rho_m 脳 V 脳 g = 13,600 kg/m^3 脳 V 脳 g\) (d) Evacuated chamber: \(F_{b_{evacuated}} = 0 kg/m^3 脳 V 脳 g = 0\)
04

Determine the relative magnitudes of the buoyancy forces

Comparing the buoyancy forces, we can see that \(F_{b_{mercury}} > F_{b_{water}} > F_{b_{air}} > F_{b_{evacuated}}\). The buoyancy force is greatest when the body is submerged in mercury, followed by water, then air, and finally, no buoyancy force is present in an evacuated chamber.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Archimedes' Principle
Archimedes' Principle provides a fundamental explanation of buoyancy. It tells us that any object submerged in a fluid experiences an upward force. This force, called the buoyancy force, is equal to the weight of the fluid displaced by the object.
In mathematics, Archimedes' Principle is expressed as:
  • \( F_b = \rho \times V \times g \)
Here, \( F_b \) is the buoyancy force, \( \rho \) is the fluid's density, \( V \) is the volume of the displaced fluid, and \( g \) is the acceleration due to gravity.
Archimedes discovered this principle while pondering why some objects float while others sink. He realized that as an object enters a fluid, it pushes aside a certain volume of that fluid. The more fluid gets displaced, the higher the upward lift will be.
Fluid Density
Fluid density plays a crucial role in determining the buoyancy force. It measures how much mass is contained in a given volume of a fluid.
Common units for density are \( kg/m^3 \). Here's how the densities for air, water, mercury, and an evacuated chamber stack up:
  • Air: \( 1.225 \text{ kg/m}^3 \)
  • Water: \( 1,000 \text{ kg/m}^3 \)
  • Mercury: \( 13,600 \text{ kg/m}^3 \)
  • Evacuated chamber: \( 0 \text{ kg/m}^3 \)
A higher density means more mass in the same volume, which increases the weight of displaced fluid and thus, the buoyancy force. This is why mercury, with its high density, offers a stronger buoyancy force than water or air.
Buoyancy Comparison
To compare the buoyancy forces in different environments, we use the formula from Archimedes' Principle: \( F_b = \rho \times V \times g \). The larger the density \( \rho \), the larger the force becomes.
For a given volume, calculating the buoyancy forces:
  • In air: Low density, results in a smaller buoyancy force.
  • In water: Higher density than air, larger buoyancy force.
  • In mercury: Very high density, hence the largest buoyancy force.
  • In an evacuated chamber: Density is zero, so no buoyancy force.
This sequence shows that the buoyancy force is strongest in mercury, followed by water, then air, and in a vacuum, it doesn't exist at all.
Educational Physics
Educational physics involves understanding principles like these to grasp how the world works. When you learn about buoyancy, you gain insights into why boats float on water or why a helium balloon rises. Archimedes' Principle is a cornerstone of this understanding, illustrating clear laws that govern everyday phenomena.
When explaining topics like buoyancy, it's important to discuss:
  • The role of fluid density
  • How volume affects displaced fluid
  • Real-world applications like designing ships and submarines
Educational physics seeks to make such concepts clear and vivid, bridging the gap between theory and practical observation.

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