91Ó°ÊÓ

Determining the density of a fluid has many imporLant applications. A car battery contains sulfuric acid, and the battery will not function properly if the acid density is too low. Similarly, the effectiveness of antifreeze in your car's engine coolant depends on the density of the mixture (usually ethylene glycol and water). When you donate blood to a blood bank, its screcning includes a determination of the density of the blood because higher density correlates with higher hemoglobin content. A hydrometer is an instrument used to determine the density of a liquid. \(A\) simple one is sketched in Figure \(\mathrm{P} 9.84\). The bulb of a syringe is squeezed and released to lift a sample of the liquid of interest into a tube containing a calibrated rod of known density. (Assume the rod is cylindrical.) The rod, of length \(L\) and average density \(\rho_{0}\). floats partially immersed in the liquid of density \(p .\) A length \(h\) of the tod protrudes above the surface of the liquid. Show that the density of the liquid is given by $$ \rho=\frac{\rho_{11} L}{I_{A}-h} $$

Step by step solution

01

Recall the Archimedes' principle of buoyancy

According to the Archimedes' principle, any object, wholly or partially immersed in a fluid, is buoyed up by a force equal to the weight of the fluid displaced by the object. The buoyant force (\(F_B\)) on the rod can therefore be expressed as \(F_B = Vf \cdot \rho \cdot g\), where \(Vf\) is the volume of fluid displaced, \(\rho\) is the density of the fluid, and \(g\) is the acceleration due to gravity.

02

Express the volume of the fluid

The volume of the liquid displaced by the rod is equal to the volume of the rod submerged in the fluid (\(Vf\)). In terms of the given parameters, this volume can be written as \(L-h\). Therefore, the buoyant force becomes \(F_B = (L-h) \cdot \rho \cdot g\).

03

Set up the equation for equilibrium

The rod floats in equilibrium, so the upward buoyant force equals to the downward weight of the rod. The latter can be expressed as \(F_g = L \cdot \rho_0 \cdot g\), where \(\rho_0\) is the density of the rod. Therefore, the equation for equilibrium is \(F_g = F_B\), or \(L \cdot \rho_0 \cdot g = (L-h) \cdot \rho \cdot g\).

04

Solve the equation for \(\rho\)

The goal is to find an expression for the density of the liquid, \(\rho\). From the equilibrium equation, it's clear that \(g\) can be canceled out. Rearrange the equation to solve for \(\rho\): \(\rho = \frac{L \cdot \rho_0}{L-h}\).

Unlock Step-by-Step Solutions & Ace Your Exams!

• Full Textbook Solutions

  Get detailed explanations and key concepts

• Unlimited Al creation

  Al flashcards, explanations, exams and more...

• Ads-free access

  To over 500 millions flashcards

• Money-back guarantee

  We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91Ó°ÊÓ!

Key Concepts

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

Density Measurement

Density is a fundamental property that helps us understand how much mass is present in a given volume of a substance. This is crucial in many practical applications. Whether it's for ensuring the right mixture of coolant in your car or verifying the strength of your blood, knowing density is essential. Density (\( \rho \)) is typically calculated using the formula:

• \( \rho = \frac{m}{V} \)

where \( m \) is the mass and \( V \) is the volume. This formula tells us how tightly matter is packed in a space.
In liquids, density can help determine purity or concentration. For example, sulfuric acid in car batteries or antifreeze solutions's effectiveness.
The measurement often involves using tools like hydrometers, which use principles of fluid mechanics to give accurate readings of liquid density.

Buoyant Force

The buoyant force is what allows objects to float or partially submerge in a liquid. Based on Archimedes' Principle, it states that the buoyant force exerted on an object submerged in a fluid is equal to the weight of the fluid displaced by the object.
The formula for buoyant force (\( F_{B} \)) is:

• \( F_{B} = V_{f} \cdot \rho \cdot g \)

where \( V_{f} \) is the displaced liquid volume, \( \rho \) is the liquid's density, and \( g \) is the acceleration due to gravity.
When an object like a rod floats, the buoyant force counteracts its weight, allowing it to remain in equilibrium. This balance between the weight of the object and the buoyant force is why boats float and hydrometers work effectively to measure liquid density.

Volume Displacement

Volume displacement occurs when a solid object is placed in a liquid, causing the liquid level to rise. This principle is crucial in measuring the volume of irregularly shaped objects and in understanding buoyancy.
In the context of a hydrometer, a rod displaces a volume of liquid as it sinks. The part of the rod submerged pushes aside a volume (\( V_{f} \)) equivalent to what it occupies. This concept is key when calculating buoyant force and thereby understanding how objects like hydrometers float.

• The displaced volume can be calculated as: \( L - h \)

where \( L \) is the total rod length and \( h \) is the protruding part above the liquid's surface. The rod's floating balance allows us to measure the liquid's density using this principle.

Hydrometer Functionality

A hydrometer is an intuitive device designed to measure the density of liquids accurately. It consists of a tube with a weighted end, allowing it to float upright in liquid. The tube often holds a calibrated scale for reading.
The hydrometer principle relies on buoyancy and volume displacement. When dipped in a liquid, it sinks until the weight of the displaced liquid equals the hydrometer's weight. The denser the liquid, the less it needs to sink to reach equilibrium, hence more of the tube remains above the liquid. This displacement relationship gives a direct reading of density.
Hydrometers are widely used in industries such as brewing, winemaking, and automotive to evaluate liquid per unit mass. Their functionality showcases a practical application of Archimedes' Principle in everyday technology.