Ch 18: A Macroscopic Description of Matter

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 18: A Macroscopic Description of Matter

Problem 49

Chapter 18, Problem 49

The 3.0-m-long pipe in FIGURE P18.49 is closed at the top end. It is slowly pushed straight down into the water until the top end of the pipe is level with the water's surface. What is the length L of the trapped volume of air?

Verified step by step guidance

Step 1: Identify the physical principles involved. This problem deals with the behavior of gases under pressure, specifically Boyle's Law, which states that for a fixed amount of gas at constant temperature, the product of pressure and volume remains constant:

P_{1} V_{1} = P_{2} V_{2}

Step 2: Define the initial conditions. Initially, the pipe is open to the atmosphere, so the pressure inside the pipe is equal to atmospheric pressure (

P_{1}

). The volume of air trapped in the pipe is the full length of the pipe multiplied by its cross-sectional area (

= A × 3.0 \(\text{ m}\)

Step 3: Define the final conditions. When the pipe is pushed into the water, the top end is level with the water's surface, and the trapped air is compressed. The pressure inside the pipe increases due to the water column above the trapped air. The final pressure (

P_{2}

) is equal to atmospheric pressure plus the pressure due to the water column (

P_{2} = P_{atm} + 
ho g h

, where

ho

is the density of water,

g

is the acceleration due to gravity, and

h

is the depth of the pipe submerged).

Step 4: Relate the final volume of air to the length of the trapped air column. The final volume of air (

V_{2}

) is equal to the cross-sectional area of the pipe multiplied by the length of the trapped air column (

V_{2} = A × L

Step 5: Apply Boyle's Law to solve for the length of the trapped air column. Using

P_{1} V_{1} = P_{2} V_{2}

, substitute the known values and expressions for

P_{1}

V_{1}

P_{2}

, and

V_{2}

to solve for

L

, the length of the trapped air column.

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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 increases with depth in a fluid and 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 depth of the fluid. Understanding hydrostatic pressure is essential for analyzing how the water level affects the trapped air in the pipe.

Ideal Gas Law

The Ideal Gas Law relates the pressure, volume, and temperature of an ideal gas through the equation PV = nRT. In this context, as the pipe is submerged, the volume of the trapped air will change due to the pressure exerted by the water above it. This relationship is crucial for determining how the length of the trapped air volume changes as the pipe is pushed down.

Buoyancy

Buoyancy is the upward force exerted by a fluid on an object submerged in it, which is equal to the weight of the fluid displaced by the object. This principle, described by Archimedes' principle, is important for understanding how the pipe interacts with the water and how the submerged length of the pipe affects the volume of air trapped inside it.

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