Problem 66 - University Physics with Modern Physics

Question

A hydrometer consists of a spherical bulb and a cylindrical stem with a cross- sectional area of 0.400 \(\mathrm{cm}^{2}\) (see Fig. 12.12 \(\mathrm{a} ) .\) The total volume of bulb and stem is 13.2 \(\mathrm{cm}^{3} .\) When immersed in water, the hydrometer floats with 8.00 cm of the stem above the water surface. When the hydrometer is immersed in an organic fluid, 3.20 cm of the stem is above the surface. Find the density of the organic fluid. (Note: This illustrates the precision of such a hydrometer. Relatively small density differences give rise to relatively large differences in hydrometer readings.)

Step-by-Step Solution

Answer

The density of the organic fluid is approximately 0.992 g/cm³.

Step 1: Understand the problem

The hydrometer is floating, so the weight of the displaced fluid is equal to the weight of the hydrometer. We need to find the density of the organic fluid that results in 3.20 cm of the stem being above the fluid surface.

Step 2: Calculate displaced volume of water

When 8.00 cm of the stem is above water, the submerged part of the stem is 12.80 cm (since the total volume is 13.2 cm³ and the stem is 8.00 cm long). The volume of the submerged part in water is 12.80 cm x 0.400 cm² = 5.12 cm³. So, the displaced volume of water when in water is 13.2 cm³ - (8 cm x 0.400 cm²) = 9.04 cm³.

Step 3: Calculate weight of the hydrometer

The weight of the hydrometer is then the weight of 9.04 cm³ of water, which is 9.04 g (since the density of water is 1 g/cm³).

Step 4: Calculate the displaced volume in the organic fluid

In the organic fluid, 3.20 cm of the stem is above the fluid, meaning that 10.00 cm is submerged. The submerged volume is 10.00 cm x 0.400 cm² + 13.2 cm³ - (8.00 cm x 0.400 cm²) = 9.12 cm³.

Step 5: Use Archimedes' principle in the organic fluid

The weight of the organic fluid displaced by the hydrometer is equal to the weight of the hydrometer, which was calculated as 9.04 g. Thus, if the displaced organic fluid volume is 9.12 cm³, the density \( \rho \) of the organic fluid can be found using \( \rho = \frac{9.04 \, \text{g}}{9.12 \, \text{cm}^3} \approx 0.992 \, \text{g/cm}^3 \).

Key Concepts

Density Calculation

Density is a fundamental concept in physics and fluid mechanics, representing the mass per unit volume of a substance. It is usually expressed in units such as grams per cubic centimeter (g/cm\(^3\)) or kilograms per cubic meter (kg/m\(^3\)).
To calculate density, use the formula: \( \text{Density} \, (\rho) = \frac{\text{Mass}}{\text{Volume}} \).
The problem with the hydrometer involves understanding how the mass of a fluid displaced relates to the volume displaced, thereby helping to determine the fluid's density.
1. **Displaced Volume**: The volume of fluid displaced equals the volume of the submerged part of the hydrometer.

• When submerged in water, 9.04 cm³ of water is displaced.
• When submerged in the organic fluid, 9.12 cm³ of fluid is displaced.

The density of the organic fluid can then be calculated by knowing that displaced fluid equals the hydrometer's weight in both cases.

Archimedes' Principle

Archimedes' Principle is pivotal in understanding why objects float or sink. It states that a body immersed in a fluid experiences an upward force, termed buoyancy, which is equivalent to the weight of the fluid it displaces.
This principle is crucial because it explains why the hydrometer floats at different levels in various fluids. The weight of the hydrometer does not change, so the difference in the floating level between water and the organic liquid gives insights about their densities.

• **Weight Equivalence**: The hydrometer's weight is balanced by the buoyant force acting upward, which is equal to the weight of the fluid displaced.
• In water: The hydrometer's weight matches the weight of 9.04 g of displaced water.
• In the organic fluid: This weight remains the same (9.04 g), but the volume displaced changes to 9.12 cm³.

Fluid Mechanics

Fluid mechanics is the branch of physics concerned with the behavior of fluids (liquids and gases) in motion and at rest. In this context, understanding the buoyancy and behavior of the hydrometer in different fluids is essential.
Fluid properties such as density and viscosity influence an object's buoyancy.

• **Buoyant Force**: As per Archimedes' Principle, fluid displaced creates an upward buoyant force, affecting how deep an object sinks.
• **Hydrometer Behavior**: The changing levels of submersion of the hydrometer stem indicate variations in density between the fluids tested.

The study of fluid mechanics helps predict and model behaviors of fluids and submerged objects under different conditions, providing a foundation for calculating unseen properties based on observable phenomena.

Density of Organic Fluids

The density of organic fluids can vary significantly based on their composition and temperature. The exercise illustrates the precision required when measuring slight differences in density using a hydrometer.
Organic fluids generally have densities less than water, reflected here by the lesser displacement needed to balance the hydrometer.

• The density calculation for the organic fluid yields approximately 0.992 g/cm³.
• **Precision Importance**: Small differences in density can significantly influence the hydrometer's indication, hence the need for precision in volumetric measurements.
• This exercise underscores the hydrometer's efficiency in distinguishing fluid densities with small variations.

Considering these variations is important, as they can affect processes where precise fluid behavior is critical, such as chemical engineering and environmental science applications.