Problem 83 - University Physics with Modern Physics

Question

A cubical block of density \(\rho_{\mathrm{B}}\) and with sides of length \(L\) floats in a liquid of greater density \(\rho_{\mathrm{L}}\) . (a) What fraction of the block's volume is above the surface of the liquid? (b) The liquid is denser than water (density \(\rho_{\mathrm{W}} )\) and does not mix with it. If water is poured on the surface of the 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 \(L, \rho_{\mathrm{B}}, \rho_{\mathrm{L}},\) and \(\rho_{\mathrm{W}}\) (c) Find the depth of the water layer in part (b) if the liquid is mercury, the block is made of iron, and the side length is 10.0 \(\mathrm{cm} .\)

Step-by-Step Solution

Answer

(a) \( 1 - \frac{\rho_{B}}{\rho_{L}} \). (b) Water depth cannot be achieved as \(h\) is negative.

Step 1: Understand Archimedes' Principle

According to Archimedes' Principle, a body submerged in a fluid experiences an upward buoyant force equal to the weight of the fluid displaced by it. For a floating object, this buoyant force equals the weight of the object itself. The block floats because the buoyant force (due to displaced liquid) balances the block's weight. Mathematically, this is \( \rho_{\mathrm{L}} g V_{\mathrm{displaced}} = \rho_{\mathrm{B}} g V_{\mathrm{total}} \), where \( V_{\mathrm{displaced}} \) is the submerged volume.

Step 2: Determine Fraction of Volume Above Liquid

The fraction of the block's volume that is submerged is equal to the density of the block divided by the density of the liquid: \( \frac{V_{\mathrm{sub}}}{V_{\mathrm{total}}} = \frac{\rho_{\mathrm{B}}}{\rho_{\mathrm{L}}} \). Therefore, the fraction of the block's volume that is above the surface is: \( 1 - \frac{\rho_{\mathrm{B}}}{\rho_{\mathrm{L}}} \).

Step 3: Situation After Adding Water

When water is poured on top, it adds an additional downward force. For the water surface to rise to the top of the block, the buoyant force together with the additional weight of the block must balance the total downward forces.

Step 4: Calculate Water Layer Depth

Let \( h \) be the depth of the water layer. The condition for the water level just reaching the top of the block is given by balancing forces: \( \rho_{\mathrm{L}} g (L-h) L^2 + \rho_{\mathrm{W}} g h L^2 = \rho_{\mathrm{B}} g L^3 \). Solving for \( h \), we get: \[ h = L \left( 1 - \frac{\rho_{\mathrm{B}}}{\rho_{\mathrm{W}}} \right) - \frac{L}{\rho_{\mathrm{W}}}\left( \rho_{\mathrm{L}} - \rho_{\mathrm{B}} \right) \].

Step 5: Numerical Calculation for Specific Materials

With mercury as the liquid (\( \rho_{\mathrm{L}} = 13534 \, \mathrm{kg/m^3} \)) and iron as the block (\( \rho_{\mathrm{B}} = 7874 \, \mathrm{kg/m^3} \)), and \( L = 0.1 \, \mathrm{m} \), \( \rho_{\mathrm{W}} = 1000 \, \mathrm{kg/m^3} \) the equation becomes: \[ h = 0.1 \left( 1 - \frac{7874}{1000} \right) - \frac{0.1}{1000} \times (13534 - 7874) \] Calculating gives \( h \approx -0.037 \, \mathrm{m} \), but since depth cannot be negative, this means water density can't support the mercury block further.

Key Concepts

Buoyant Force

The concept of buoyant force is central to understanding why objects float or sink in fluid mechanics. According to Archimedes' Principle, an object submerged in a fluid experiences an upward force known as the buoyant force. This force is equivalent to the weight of the fluid displaced by the object. For floating objects, the buoyant force balances the object's weight, allowing it to stay at the surface.
In mathematical terms, if a block is floating in a liquid, the buoyant force can be described as:

• Buoyant Force = Weight of Displaced Fluid

If this force equals the block's weight, the block floats. This is crucial for part (a) of the exercise, where it is necessary to understand that the fraction of the block submerged is related to its density relative to the liquid it's in.

Density

Density is a key factor in determining whether an object will sink or float in a fluid. Density describes how much mass an object has in a given volume, and it is expressed as mass per unit volume (e.g., kg/m³).
Here's how density affects floating:

• If an object has a lower density than the fluid, it will float.
• If the density is higher, it will sink.
• If the densities are equal, the object will hover in the fluid.

In the original exercise, the block has a density of (\( \rho_{\mathrm{B}} \)) and floats on a liquid with a greater density (\( \rho_{\mathrm{L}} \)). The fraction of the block submerged is given by the ratio \( \frac{\rho_{\mathrm{B}}}{\rho_{\mathrm{L}}} \), meaning the block's density compared to the liquid determines how much of it will be above the surface.

Floating Objects

When objects float, it means that their weight is fully supported by the upward buoyant force exerted by the fluid. This floating equilibrium arises from a balance of forces, and it's why certain items remain at the surface while others do not.
The exercise illustrates how a block can float due to its specific density compared to the liquid, emphasizing the significance of its volume that remains above the fluid surface.

• Fraction of Volume Above Surface = (\( \frac{V_{\mathrm{total}} - V_{\mathrm{sub}}}{V_{\mathrm{total}}} \))

This fraction can be calculated using the ratio of densities as seen in the exercise. Floating objects also show how addition of a liquid layer, as in part (b), becomes complex when dealing with fluids of varying densities.

Fluid Mechanics

Fluid mechanics is a branch of physics that studies how fluids behave, both in motion and at rest. It encompasses the principles behind buoyancy, density, and the behavior of gases and liquids. In this context, understanding fluid dynamics helps explain the interaction between the block, liquid, and the additional water layer added in the exercise.
Key principles in fluid mechanics include:

• Fluid Statics: the study of fluids at rest.
• Fluid Dynamics: the study of fluids in motion.

In the scenario from the exercise, the principles of fluid statics are used to determine the stability of the floating block and the balance of additional forces introduced by the water layer, ensuring the overall mechanical equilibrium within the system.