Ch 14: Fluids and Elasticity

Knight Calc - Physics for Scientists and Engineers 5th Edition

Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Not the one you use?Change textbook

Knight Calc 5th Edition

Ch 14: Fluids and Elasticity

Problem 46

Chapter 14, Problem 46

Glycerin is poured into an open U-shaped tube until the height in both sides is 20 cm. Ethyl alcohol is then poured into one arm until the height of the alcohol column is 20 cm. The two liquids do not mix. What is the difference in height between the top surface of the glycerin and the top surface of the alcohol?

Verified step by step guidance

Step 1: Understand the problem setup. The U-shaped tube contains two immiscible liquids: glycerin and ethyl alcohol. The glycerin is initially at the same height (20 cm) in both arms of the tube. Ethyl alcohol is then added to one arm, creating a column of 20 cm. The goal is to find the difference in height between the top surfaces of glycerin and alcohol.

Step 2: Recall the principle of hydrostatic equilibrium. In a U-shaped tube, the pressure at the same horizontal level in both arms must be equal. The pressure at the bottom of the glycerin column in one arm must equal the pressure at the bottom of the alcohol column plus the glycerin column in the other arm.

Step 3: Write the pressure equations for both arms. The pressure at the bottom of the glycerin column is given by \( P_{glycerin} = \rho_{glycerin} g h_{glycerin} \), where \( \rho_{glycerin} \) is the density of glycerin, \( g \) is the acceleration due to gravity, and \( h_{glycerin} \) is the height of the glycerin column. For the arm with alcohol, the pressure is \( P_{alcohol+glycerin} = \rho_{alcohol} g h_{alcohol} + \rho_{glycerin} g h_{glycerin} \).

Step 4: Set the pressures equal to each other. Since the pressures at the bottom of both arms are equal, \( \rho_{glycerin} g h_{glycerin} = \rho_{alcohol} g h_{alcohol} + \rho_{glycerin} g h_{glycerin}' \), where \( h_{glycerin}' \) is the new height of glycerin in the arm with alcohol.

Step 5: Solve for the height difference. Rearrange the equation to find \( h_{glycerin} - h_{glycerin}' \), which represents the difference in height between the top surface of glycerin and the top surface of alcohol. Use the known densities of glycerin and ethyl alcohol to calculate the result.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Hydrostatic Pressure

Hydrostatic pressure is the pressure exerted by a fluid at equilibrium due to the force of gravity. It is given by the formula P = ρgh, where P is the pressure, ρ is the fluid density, g is the acceleration due to gravity, and h is the height of the fluid column. In this scenario, the pressure exerted by the glycerin and ethyl alcohol columns will determine the height difference between their surfaces.

Density of Liquids

Density is a measure of mass per unit volume of a substance, typically expressed in grams per cubic centimeter (g/cm³) for liquids. Glycerin has a higher density (approximately 1.26 g/cm³) compared to ethyl alcohol (approximately 0.789 g/cm³). This difference in density is crucial for calculating the height difference between the two liquid surfaces in the U-tube, as it affects the hydrostatic pressure exerted by each liquid.

Equilibrium in Fluid Systems

In a fluid system, equilibrium occurs when the pressures exerted by different fluid columns balance each other out. In the U-tube scenario, the heights of the glycerin and ethyl alcohol columns will adjust until the pressures at the same horizontal level are equal. This principle allows us to determine the height difference between the two liquid surfaces based on their respective densities and the heights of the columns.

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