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 submergcd. 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 densitics relative to that of water) of (a) \(A,(b) B,\) and \((c) C ?\)

Step-by-Step Solution

Answer

(a) 0.5, (b) 0.333, (c) 0.75

Step 1: Understand Buoyancy in Water

Since the block is fully submerged in water and floats, its density is equal to the density of water. Let the density of the block be denoted as \( \rho_{\text{block}} \) and the density of water be \( \rho_{\text{water}} \). Thus, \( \rho_{\text{block}} = \rho_{\text{water}} \).

Step 2: Setup for Liquid A

In liquid A, the block floats with a height \( h/2 \) above the surface. This means \( h/2 \) of the block is submerged. According to Archimedes' Principle, the volume submerged \( V_{\text{sub}} \) determines the buoyancy force, thus \( V_{\text{sub}} = \frac{h}{2} \).

Step 3: Calculate Relative Density of Liquid A

Since \( \rho_{\text{block}} = \rho_{\text{water}} \), and using fractions for the submerged part conforms to the principle: \( \frac{\rho_{A}}{\rho_{\text{block}}} = \frac{V_{\text{sub}}}{h} \). Thus, \( \rho_{A} = \frac{1}{2} \rho_{\text{water}} \).

Step 4: Setup for Liquid B

In liquid B, the block floats with a height \( 2h/3 \) above the surface, which means \( h/3 \) is submerged in the liquid. The volume submerged is \( \frac{h}{3} \).

Step 5: Calculate Relative Density of Liquid B

The density ratio for liquid B is: \( \frac{\rho_{B}}{\rho_{\text{block}}} = \frac{V_{\text{sub}}}{h} \). Thus, \( \rho_{B} = \frac{1}{3} \rho_{\text{water}} \).

Step 6: Setup for Liquid C

In liquid C, the block floats with height \( h/4 \) above the liquid surface. This means \( 3h/4 \) is submerged, giving \( V_{\text{sub}} = \frac{3h}{4} \).

Step 7: Calculate Relative Density of Liquid C

For liquid C, follow the same principle: \( \frac{\rho_{C}}{\rho_{\text{block}}} = \frac{V_{\text{sub}}}{h} \). Thus, \( \rho_{C} = \frac{3}{4} \rho_{\text{water}} \).

Key Concepts

Archimedes' Principle

Archimedes' Principle is a fundamental concept that helps us understand why objects float or sink in a fluid. It states that any object submerged in a fluid is buoyed up by a force equal to the weight of the fluid displaced by that object. This principle is crucial in the exercise because it explains the floating behavior of the block in different liquids.

When the block is submerged, it displaces a certain volume of liquid, and the buoyancy force acting upwards is equal to the weight of this displaced liquid. If the buoyancy force is equal to the weight of the block, the block floats. If the block weighs more, it will sink; if it weighs less, it will rise. This balance determines the extent of the block's submersion in each fluid.

Understanding this principle allows us to predict and calculate how much of the block stays submerged in various liquids and to assess their relative densities.

Density

Density is a measure of how much mass is contained in a given volume. It is often expressed in units like kilograms per cubic meter (kg/m³) or grams per cubic centimeter (g/cm³). In this exercise, the density of the block is equal to the density of water since it floats fully submerged in water.

Mathematically, density is defined by the formula: \( \rho = \frac{m}{V} \) where \( \rho \) is density, \( m \) is mass, and \( V \) is volume. The density of an object determines whether it will float or sink when placed in a fluid. A denser object will sink, and a less dense object will float.

By understanding the density of both the fluid and the object, we can predict the object's behavior when submerged. In liquids A, B, and C, knowing the density allows us to determine how much of the block is above or below the surface, showcasing its role in buoyancy.

Relative Density

Relative density, also called specific gravity, compares the density of a substance with the density of a reference substance, typically water for liquids. It is a dimensionless quantity because it is a ratio.

For example, a relative density less than 1 indicates that the substance is less dense than water, and as such, the substance will float if it’s a liquid. In this exercise, we calculate the relative densities of liquids A, B, and C using the proportion of the block's submerged volume to relate back to the density of water.

The calculations are done as follows:

• Liquid A: Relative density is \( \frac{1}{2} \), making it less dense than water.
• Liquid B: Relative density is \( \frac{1}{3} \), also less dense than water.
• Liquid C: Relative density is \( \frac{3}{4} \), meaning it is denser than both A and B, but still less dense than water.

Floating and Submerged Volumes

Understanding floating and submerged volumes involves knowing how much of an object's volume stays beneath the liquid's surface. This is important for solving how the block floats in different liquids.

The submerged volume in water equates to the entire volume of the block because it floats fully submerged. In other liquids:

• For liquid A: The block floats with half the height submerged \( \frac{h}{2} \), indicating a smaller displaced volume compared to water.
• For liquid B: A third of the block is submerged \( \frac{h}{3} \), displacing less liquid than in A.
• For liquid C: Here, three-quarters of the block is submerged \( \frac{3h}{4} \), displacing more liquid than in A or B but still less than in full submersion.

These differences in submerged volumes are directly tied to the fluid's density and its ability to support the weight of the block, illustrating how floating behavior changes with the relative density of each liquid.