Problem 80 - Fundamentals of Physics

Question

In an experiment, a rectangular block with height \(h\) is allowed to float in four separate liquids. In the first liquid, which is water, it floats fully submerged. In liquids \(A, B\), and \(C\), it floats with heights \(h / 2,2 h / 3\), and \(h / 4\) above the liquid surface, respectively. What are the relative densities (the densities relative to that of water) of (a) \(A,(\mathrm{~b}) B\), and \((\mathrm{c}) C ?\)

Step-by-Step Solution

Answer

(a) 0.5, (b) 0.33, (c) 0.75

Step 1: Understanding the Problem

In this exercise, we are given that a block floats fully submerged in water, meaning its density equals that of water. When floating in liquids A, B, and C, only certain portions of the block's height are submerged, indicating different densities compared to water. The goal is to find the relative densities of the liquids compared to water.

Step 1: Setup the Density Relation for Liquid A

For Liquid A, the block floats with height \(h/2\) above the surface, indicating \(h/2\) is submerged. The block's submerged volume in A is \(h/2\), suggesting the buoyant force equals the block's weight. The relative density of A with respect to water is the fraction submerged: \(\rho_A = \frac{\text{submerged height}}{h} = \frac{h/2}{h} = \frac{1}{2}\). Thus, \(\rho_A = 0.5\).

Step 2: Setup the Density Relation for Liquid B

For Liquid B, the block floats with height \(2h/3\) above the surface; hence, \(h/3\) is submerged. Executing a similar calculation gives \(\rho_B = \frac{\text{submerged height}}{h} = \frac{h/3}{h} = \frac{1}{3}\). Thus, \(\rho_B = 0.33\bar{3}\).

Step 3: Setup the Density Relation for Liquid C

For Liquid C, only \(h/4\) is above the surface, with \(3h/4\) submerged. Following the same process, we find \(\rho_C = \frac{\text{submerged height}}{h} = \frac{3h/4}{h} = \frac{3}{4}\). Thus, \(\rho_C = 0.75\).

Step 5: Final Calculation

From the above calculations, we determine the relative densities of the liquids A, B, and C with respect to water.

Key Concepts

Relative density

Relative density, also known as specific gravity, is a measure of the density of a substance compared to the density of a reference substance, usually water. In simple terms, it's a comparison of how dense one fluid is to another. This concept helps us understand how objects interact with different liquids, predicting whether they will sink or float.

• Definition: Relative density is the density of an object divided by the density of water (or another reference fluid).
• Mathematical Expression: \( \text{Relative Density} = \frac{\text{Density of Object}}{\text{Density of Water}} \).

For this problem, the relative density tells us how much of the block will be submerged when it floats in different liquids. This is because the denser the liquid, the less of the block needs to be submerged to displace enough liquid to balance its weight.
The relative densities in this scenario are the submerged portions of the block, which provide an intuitive grasp of the interaction between the block and each specific liquid.

Archimedes' principle

Archimedes' principle is essential to understanding buoyancy and how objects float. This principle states that an object submerged in a fluid experiences an upward force, known as the buoyant force. This force is equal to the weight of the fluid that is displaced by the object. It's a foundational concept in fluid mechanics.

• Buoyant Force: The buoyant force acts in the upward direction, counteracting the weight of the object.
• Mathematical Expression: \( F_b = \text{weight of displaced fluid} = V \cdot \rho_f \cdot g \).

Here, \(V\) is the volume of the fluid displaced, \(\rho_f\) is the fluid's density, and \(g\) is the acceleration due to gravity.
In the context of this problem, Archimedes’ principle explains why only part of the block is submerged in each liquid, except water where it is fully submerged. The block's weight is balanced by the buoyant force, which varies in each liquid due to their differing densities.

Floating objects

When we talk about floating objects, we're referring to objects that remain partially or wholly on top of a fluid due to buoyancy. The concept of floating depends on the densities of both the object and the fluid in which it is placed.

• Condition for Floating: An object will float if its density is lower than the fluid, resulting in sufficient buoyant force to counteract its weight.
• Mathematical Insight: For an object to float with a certain fraction of its height above the liquid, the buoyant force must equal the gravitational pull on the entire volume of the object. Mathematically, \( \rho_s \times V_s \times g = \rho_f \times V_f \times g \), where \(s\) indicates object and \(f\) indicates fluid.

This problem illustrates floating as it shows different levels of submergence of the block in various liquids. The height of the block above the liquid's surface gives us a direct measure of the fluid's relative density to water. As seen in the problem, the block floats differently in each fluid due to the changes in the density of the liquids.