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Chapter 12: Problem 75

A cubical block of density \({\rho_B}\) and with sides of length \(L\) floats in a liquid of greater density \({\rho_L}\). (a) What fraction of the block's volume is above the surface of the liquid? (b) The liquid is denser than water (density \({\rho_W}\)) and does not mix with it. If water is poured on the surface of that liquid, how deep must the water layer be so that the water surface just rises to the top of the block? Express your answer in terms of \(L\), \({\rho_B}\), \({\rho_L}\), and \({\rho_W}\). (c) Find the depth of the water layer in part (b) if the liquid is mercury, the block is made of iron, and \(L\) \(=\) 10.0 cm.

Short Answer

Expert verified
(a) Fraction above surface: \( 1 - \frac{\rho_B}{\rho_L} \). (b) Water depth \( h_w = \frac{L(1 - \frac{\rho_B}{\rho_L})\rho_L}{\rho_W} \). (c) 4.8 cm.

Step by step solution

01

Understand the Concept of Buoyancy

When an object floats, the weight of the fluid displaced by the submerged portion of the object is equal to the weight of the object itself. The principle that governs this is Archimedes' principle.
02

Calculate the Submerged Volume Fraction

Let's denote the fraction of the block's volume that is submerged as \( x \). According to Archimedes' principle, the buoyant force is equal to the weight of the block:\[ \text{Buoyant force} = \rho_L \cdot g \cdot (L^3 \cdot x) = \rho_B \cdot g \cdot L^3 \]Where \( L^3 \) is the volume of the cube. Solving for \( x \), we have:\[ x = \frac{\rho_B}{\rho_L} \]The fraction above the liquid is \( 1 - x \).
03

Find the Fraction Above the Liquid Surface (Part a)

From the expression for \( x \) in Step 2:\[ 1 - x = 1 - \frac{\rho_B}{\rho_L} \] This represents the fraction of the block's volume that is above the surface of the liquid.
04

Establish Condition for Second Liquid Layer (Part b)

When water is added on top of the denser liquid, the block must still float, this time with the water surface reaching just the top of the block. The additional buoyant force from the water is considered. Let \( h_w \) be the depth of water:\[ \rho_L \cdot g \cdot (xL^3) + \rho_W \cdot g \cdot (h_w \cdot L^2) = \rho_B \cdot g \cdot L^3 \]Solving for \( h_w \):\[ h_w = \frac{L(1-x)\rho_L}{\rho_W} \approx \frac{L\left(1 - \frac{\rho_B}{\rho_L} \right)\rho_L}{\rho_W} \]
05

Substitute and Solve for Specific Values (Part c)

For mercury \( \rho_L = 13600 \, \text{kg/m}^3 \) and iron \( \rho_B = 7870 \, \text{kg/m}^3 \) with \( L = 10 \, \text{cm} = 0.1 \, \text{m} \),\( \rho_W = 1000 \, \text{kg/m}^3 \). Use the equation from Step 4:\[ h_w = \frac{0.1(1 - \frac{7870}{13600}) \cdot 13600}{1000} \approx 0.048 \text{ m} \] or 4.8 cm.

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Key Concepts

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

Buoyancy
Buoyancy is a force exerted by a fluid that opposes the weight of an object immersed in it. This force is what allows objects to float or sustain themselves within fluids such as water or air. The principle governing buoyancy is known as Archimedes' principle. This principle states that the buoyant force on an object is equal to the weight of the fluid that the object displaces when it is fully or partially submerged.
  • If an object is denser than the fluid, it will sink, as the weight of the object exceeds the buoyant force.
  • Conversely, if the object is less dense, it will float, as the buoyant force is greater than the object's weight.
When solving problems involving floating objects, the balance between gravitational force (weight) and buoyant force is usually the key consideration.
Density
Density is a measure of how much mass is contained in a given volume. It is usually denoted by the Greek letter \(\rho\), and is calculated using the formula \( \rho = \frac{m}{V} \), where \( m \) is mass and \( V \) is volume. Density plays a critical role in determining whether an object will sink or float in a fluid.
  • Objects with higher density than the fluid will typically sink.
  • Objects with lower density will float, with a portion of the object submerged equal to the ratio of the object鈥檚 density to the fluid鈥檚 density.
This relationship is evident in problems involving buoyancy, where the density of the object compared to the fluid determines how much of the object will be above the surface. Understanding density is essential for solving exercises that involve different fluids layered on top of each other.
Fluid Dynamics
Fluid dynamics is the study of how fluids (liquids and gases) flow and the forces that affect them. This field of physics applies to situations like our exercise, where fluids interact with solid objects and each other. Some basic principles of fluid dynamics can be observed whenever an object floats or changes its position in a fluid. Important principles include:
  • Viscosity, which measures a fluid's resistance to deformation or flow.
  • The concept of laminar and turbulent flow, explaining the smoothness or turbulence of fluid motion.
  • Hydrostatic pressure, which is the pressure a fluid exerts due to the force of gravity acting on it.
These principles are essential for understanding how multiple fluids, like water and mercury in the exercise, can affect floating objects differently.
Volume Displacement
Volume displacement is a key concept when dealing with buoyancy. It refers to the volume of fluid that is "pushed aside" or displaced by an object when it is submerged. According to Archimedes' principle, this displaced volume corresponds directly to the buoyant force experienced by the object.
  • The weight of the displaced fluid equals the buoyant force.
  • This explains why objects with a large volume but low density displace more fluid and have a strong buoyant force keeping them afloat.
In practical terms, calculating the displaced volume helps determine how much of an object remains above the fluid鈥檚 surface, as well as the level at which multiple fluid layers interact with the object. Mastering the concept of volume displacement provides the ability to tackle complex problems involving different fluids and floating objects.

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Most popular questions from this chapter

You are preparing some apparatus for a visit to a newly discovered planet Caasi having oceans of glycerine and a surface acceleration due to gravity of 5.40 m/s\(^2\). If your apparatus floats in the oceans on earth with 25.0% of its volume submerged, what percentage will be submerged in the glycerine oceans of Caasi?

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