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 62

Chapter 14, Problem 62

Air flows through the tube shown in FIGURE P14.62 at a rate of 1200 cm³/s. Assume that air is an ideal fluid. What is the height h of mercury in the right side of the U-tube?

Verified step by step guidance

Step 1: Apply the principle of conservation of mass (continuity equation) to the air flow through the tube. The continuity equation states that the mass flow rate is constant, so the product of the cross-sectional area and velocity at one point equals the product at another point. Use the formula:

A v = A v

, where

A

is the cross-sectional area and

v

is the velocity.

Step 2: Calculate the cross-sectional areas of the two sections of the tube using the formula for the area of a circle:

A = π r^{2}

. For the first section, the radius is 1.5 mm (convert to cm: 0.15 cm), and for the second section, the radius is 3.5 cm.

Step 3: Use the continuity equation to find the velocity of air in each section of the tube. The flow rate is given as 1200 cm³/s, so the velocity can be calculated using

v = \frac{Q}{A}

, where

Q

is the flow rate and

A

is the cross-sectional area.

Step 4: Apply Bernoulli's equation to relate the pressures in the two sections of the tube. Bernoulli's equation is given as:

P + \frac{1}{2} ρ v^{2} = P + \frac{1}{2} ρ v^{2}

, where

P

is pressure,

ρ

is the density of air, and

v

is velocity.

Step 5: Relate the pressure difference to the height difference in the mercury column using the hydrostatic pressure formula:

ΔP = ρ g h

, where

ΔP

is the pressure difference,

ρ

is the density of mercury,

g

is the acceleration due to gravity, and

h

is the height difference. Solve for

h

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Key Concepts

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

Bernoulli's Principle

Bernoulli's Principle states that in a flowing fluid, an increase in the fluid's speed occurs simultaneously with a decrease in pressure or potential energy. This principle is crucial for understanding how the velocity of air flowing through the tube affects the pressure difference between the two sides of the U-tube, ultimately influencing the height of the mercury column.

Continuity Equation

The Continuity Equation is based on the principle of conservation of mass, which states that for an incompressible fluid, the mass flow rate must remain constant from one cross-section of a tube to another. In this scenario, it helps determine the relationship between the cross-sectional areas and velocities of air at different points in the tube, which is essential for calculating the pressure difference.

Hydrostatic Pressure

Hydrostatic Pressure refers to the pressure exerted by a fluid at rest due to the weight of the fluid above it. In the context of the U-tube, the height difference of the mercury column (Δh) is directly related to the hydrostatic pressure created by the air flow, allowing us to calculate the height of mercury based on the pressure difference established by the flowing air.

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