Chapter 14: Fluid Mechanics Lecture Objectives 1. Relate density, specific gravity, mass and volume; pressure, area and force; pressure, density and depth. 2. Apply Pascal’s principle in analyzing fluids in various systems

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Density – mass per unit volume

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𝝆 =𝒎

𝑽

SI unit: kg/m3

Constant for every substance

Any substance that can flow Liquids and gases Not rigid bodies Fluid dynamics from fluid

statics

Fluid

Specific gravity

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Ratio of a material’s density to the density of water (4.0°C)

𝜌𝑤𝑎𝑡𝑒𝑟,4.0℃ = 1000 kg/m3 = 1𝑔/𝑐𝑚3

• Specific gravity is a ratio of the density of a material to

the density of water.

• Specific gravity has no units.

Example

The density of gold is 19.3 g/cm3. What is its specific gravity?

Answer:

𝜌𝑎𝑖𝑟 = 1.20𝑘𝑔/𝑚3

𝜌𝑤𝑎𝑡𝑒𝑟 = 1000𝑘𝑔/𝑚3

𝝆 =𝒎

𝑽

𝐰 = 𝒎𝒈 4

Sample Problem: Weight of a roomful of air Find the mass and weight of the air in a living room 20oC with a 4.0m x 5.0 m floor and the ceiling 3.0m high. What are the mass and weight of an equal volume of water?

Given: A = 4.0m X 5.0m H = 3.0m T = 20oC Required: Mass and weight of air Mass and weight of water

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The volume of the room is V = (3.0m)(4.0m)(5.0m) = 60m3. Therefore the mass and weight of the air is:

The mass and weight of the water with the same volume is:

The density of water is a thousand time that of air; so the weight of water with the same volume is also a thousand times that of water.

Fluid pressure

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Fluid at rest exerts force perpendicular to any surface in contact with it.

If pressure is constant throughout the area,

𝒑 =𝑭⊥

𝑨

SI Unit: 1 pascal = 1 Pa = 1 N/m2

Hydrostatic equilibrium

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Net force on the fluid is zero Equal pressure on opposite directions.

Pressure difference - net force (not in equilibrium)

in Hydrostatic Equilibrium

NOT in Hydrostatic Equilibrium

Atmospheric pressure, 𝒑𝒂𝒕𝒎 Example of compressive fluid pressure; due to Earth’s atmosphere Roughly constant near the surface of the Earth

𝒑𝒂𝒕𝒎 = 𝟏 atm = 𝟏. 𝟎𝟏𝟑 bar = 𝟏. 𝟎𝟏𝟑 × 𝟏𝟎𝟓 Pa

𝒑 =𝑭┴

𝑨

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Sample problem: Force of air In a same room as the previous example (A = 4.0 x 5.0m), what is the total downward force on the surface of the floor due to air pressure of 1.00atm?

𝑭┴ = 𝒑𝑨

1atm = 1.013x105N/m2

Solution: Floor area A = 20m2; with 1atm = 1.013x105N/m2. So the total downward force is

230tons is too large! (around 30 elephants!) But the floor does not collapse because there is an upward force of equal magnitude on the underside of the floor

http://eofdreams.com/data_images/dreams/elephant/elephant-02.jpg

Elephant ~ 7-8tons

Pressure dependence

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Pressure varies with position(depth) Assume

𝝆 is constant 𝒈 is constant fluid is in equilibrium

𝒑 = 𝒑𝟎 + 𝝆𝒈𝒉

Gauge pressure is excess pressure above atmosphere pressure:

𝒑𝒈𝒂𝒖𝒈𝒆 = 𝒑 − 𝒑𝒂𝒕𝒎 = 𝒑 − 𝒑𝟎

surface pressure

The liquid in the first

container is twice as

deep, so the pressure

on the bottom is twice

that in the second

container.

Two blocks exert twice

as much pressure on

the table.

Example: Pressure dependence

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The pressure of the liquid is the same at any given depth below the

surface, regardless of the shape of the container.

What can you notice about the liquid escaping through the

hole? At which point is the “escape speed” faster? Why?

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When the liquid is pressing against a surface, there is a

force from the liquid directed perpendicular to the surface.

If there is a hole in the surface, the liquid initially will move

perpendicular to the surface.

Gravity causes the path of the liquid to curve downward.

At greater depths, the net force is greater, and the

velocity of the escaping liquid is greater.

At greater depth (or h), greater force, greater pressure

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14 http://en.wikipedia.org/wiki/Pascal%27s_barrel

Pascal's barrel - hydrostatics experiment performed by Blaise Pascal in 1646.

Pascal inserted a 10-m long vertical tube into a barrel filled with water.

What will happen when water was poured into the vertical tube?

15 http://en.wikipedia.org/wiki/Pascal%27s_barrel

Pascal's barrel - hydrostatics experiment performed by Blaise Pascal in 1646.

Pascal inserted a 10-m long vertical tube into a barrel filled with water.

After just 0.5L of water, the hydrostatic pressure increases causing the barrel to burst!

Pascal’s law

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Fluid is confined in a container. Fluid is not flowing.

𝒑 = 𝒑𝟎 + 𝝆𝒈𝒉 If we increase 𝒑𝟎 by some

amount, 𝒑 will increase exactly the same amount.

Pascal’s law states that changes in pressure at any point in an enclosed fluid at rest are transmitted undiminished to all points in the fluid and act in all directions.

http://tl.wikipedia.org/wiki/Blaise_Pascal

The force exerted on the left

piston increases the pressure

in the liquid and is transmitted

to the right piston.

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Sample problem: A storage tank 12.0m deep is filled with water. The top of the tank is open to air. What is the absolute pressure at the bottom of the tank? What is the gauge pressure?

Solution: The pressure at the bottom using; h = 12.0m; p0 = 1atm (open top) and ρ = 1000kg/m3 for water:

The gauge pressure is:

Application of Pascal’s law

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Pressure applied to an enclosed fluid is transmitted undiminished to every portion of the fluid and walls of the containing vessel.

Hydraulic Lift

𝒑 =𝑭𝟏

𝑨𝟏=

𝑭𝟐

𝑨𝟐

𝑭𝟐 =𝑨𝟐

𝑨𝟏𝑭𝟏

If A 1-N load is applied on the left piston, how much can the

piston in the right carry if the ratio of diameter of the left and right

piston is 1:50 cm2?

Example: Pascal’s law

If A 1-N load is applied on the left piston, how much can the

piston in the right carry if the diameter of the pistons are 1:50?

Example: Pascal’s law

The piston on the left has an

area of 1 cm2, and the piston

on the right has an area 50

times as great, 50 cm2.

The additional pressure of 1

N/cm2 is exerted against every

square centimeter of the larger

piston.

The larger piston will support a 50-N load - 50 times the load on the smaller piston!

𝐹𝐿

𝐴𝐿=

𝐹𝑅

𝐴𝑅→ 𝐹𝑅 =

𝐹𝐿

𝐴𝐿𝐴𝑅

=(1𝑁)(50𝑐𝑚2)

1𝑐𝑚2= 𝟓𝟎𝑵

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𝝆 =𝒎

𝑽

𝒑 =𝑭⊥

𝑨

𝒑 = 𝒑𝟎 + 𝝆𝒈𝒉

𝒑𝒈 = 𝒑 − 𝒑𝒂𝒕𝒎 = 𝒑 − 𝒑𝟎

density

pressure

(absolute) pressure

gauge pressure

𝑭𝟏

𝑨𝟏=

𝑭𝟐

𝑨𝟐

Pascal’s law (hydraulics)

𝑷𝟏 = 𝑷𝟐

Summary: Physical quantities on fluid statics

Seatwork - solve problems in your notebooks - write the answers only in your bluebook - indicate the date

January 6, 2014 1. Blah? 2. Blah blah! 3. Blah blah blah! 4. Blah blah blah blah!

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2. A hydraulic lift has two circular pistons of unequal area. A force is applied downward and perpendicular to the smaller piston making the hydraulic jack lift an object resting on the larger piston. The pressure on the larger piston is _______ the pressure on the smaller piston, and the force on the larger piston is _______ the force on the smaller piston. a. Greater than; equal to b. Greater than; greater than c. Equal to; greater than d. Equal to; less than

1. In which of the following situations would you feel the most pressure? a. In the outer space without a suit b. Inside this room while answering this seatwork c. First Physics 71 student on top of Mt. Everest d. Getting ready to jump from the top of Vinzons Hall

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p0 = pa = 980millibar = 9.80 x 104Pa ρHg = 13.6 x 103kg/m3

(absolute)p = p0+ ρgh pguage = p - p0 = ρgh

The liquid in the open tube manometer as shown is mercury; y1 = 3.00cm and y2 = 7.00cm. The atmospheric pressure pa = 980millibars.

(3) What is the absolute pressure at the bottom of the U-shaped tube (at y2)? (4) What is the absolute pressure in the open tube at a depth 4.00cm below the free surface? (5) What is the absolute pressure in the gas tank? (use h = y2-y1) (6) What is the gauge pressure of the gas in Pascals?

Answers

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2. A hydraulic lift has two circular pistons of unequal area. A force is applied downward and perpendicular to the smaller piston making the hydraulic jack lift an object resting on the larger piston. The pressure on the larger piston is _______ the pressure on the smaller piston, and the force on the larger piston is _______ the force on the smaller piston. a. Greater than; equal to b. Greater than; greater than c. Equal to; greater than d. Equal to; less than

1. In which of the following situations would you feel the most pressure? a. In the outer space without a suit b. Inside this room while answering this seatwork c. First Physics 71 student on top of Mt. Everest d. Getting ready to jump from the top of Vinzons Hall