Chapter 15: Fluids

Volume

Density

Pressure

Buoyancy

Floating

Archimedes Principle

Pascal’s Principle

Fluid Flow

Bernoulli’s Principle

Statics

Dynamics

States of Matter Solid Liquid Gas

What is the most common state of matter in the Universe?

Fluids: Liquids & Gases

•Fluids are substances that are free to flow.

•Atoms and molecules are free to move.

•They take the shape of their containers.

•Cannot withstand or exert shearing forces.

Liquids: Incompressible (density constant)

Gases: Compressible (density depends on pressure)

Parameters to describe Fluids:

Density: = mass/volume

Pressure: P = Force/Area

[P] = N/m2 = 1 Pascal (Pa)

Liquid Units

There are 1000 liters in 1 cubic meter!

1 liter = 10-3 m3 = 103 cm3

1 liter of water has a mass of 1 kg and a weight of 9.8N.

2 0 3

1 1000H

kg kg

liter m

Density • Density of water @4°C:

water = 1g/cm3 = 1000 kg/m3 = 1kg/liter

• Density of air @ 0°C:

Air = 1.29x10-3 g/cm3 = 1.29 kg/m3

Density depends on temperature! Most substances EXPAND upon heating.

m

V

How does that change their densities?

REDUCES DENSITY! m

V

m V

Water: The Exception

• Water @4°C: water =1000 kg/m3

• Ice @ 0°C: ice = 917 kg/m3

Pressure in a fluid is due to the weight

of a fluid. Force

PArea

mg

A

Pressure depends on Depth!

( )V g

A

( )Ah g

A

P gh

Pressure Acts ONLY

Perpendicularly to the Surface

Pressure depends on depth.

Pressure IN a Fluid

•Is due to the weight of the fluid above you

•Depends on Depth and Density Only

•Does NOT depend on how much water is present

•Acts perpendicular to surfaces (no shearing)

•Pressure’s add

•At a particular depth, pressure is exerted equally in ALL directions

including sideways (empirical fact)

Pressure Problem

What is the water pressure 15 m

below the surface of the lake?

Assume it is pure water.

P gh

3 21000 / (9.8 / )15kg m m s m

5 21.47 10 /x N m

147kPa 21 Pascal 1 /N m

Pressure Units

What is the water pressure 150 m

below the surface of the lake?

Assume it is pure water.

P gh

3 21000 / (9.8 / )150kg m m s m

21.47 /N m

Pressure ON a Fluid

Liquids cannot be compressed to a smaller volume.

Liquids are incompressible.

Gases can be compressed to a smaller volume.

Gases are compressible.

The Atmosphere

At sea level,

the atmosphere

has a density of

about 1.29 kg/m3.

The average

density up to

120 km is about

8.59 x10-2 kg/m3.

The Atmosphere

A square meter

extending up through

the atmosphere has a

mass of about

10,000 kg and a weight

of about 100,000 N.

1 N/m2 is a Pascal.

51 1.013 10 14.7atm x Pa psi

Measuring Pressure 51 1.013 10atm x Pa

760h mm

13.6mercury water

mercuryP gh

mercury

Ph

g

2

3 2

101,300 /

13,600 / 9.8 /

N mh

kg m x m s

P gh

Why is the pressure at X equal to atmospheric pressure?

Because if it didn’t, the mercury would

be pushed out of the dish!

31000 /water kg m

Measuring Pressure

Can a barometer be made with Water instead of Mercury?

waterP gh

water

Ph

g

2

3 2

101,300 /

1000 / 9.8 /

N mh

kg m x m s

10.3h m

(Notice: 10.3m is just 13.6 x 760mm!)

13.6mercury water

31000 /water kg m

10.3m

Mercury Barometer Water Barometer

Not to Scale!!!

51 1.013 10atm x Pa 760mm

Barometers

Measuring Air Pressure Fluid in the tube adjusts until the weight of the fluid column

balances the atmospheric force exerted on the reservoir.

Absolute vs. Gauge Pressure

• Guage pressure is

what you measure in

your tires

• Absoulte pressure is

the pressure at B and

is what is used in

PV = nRT

0Guage Pressure: P gh

0Absolute Pressure: P P gh

Measuring Pressure

absolute guage atmosphereP P P

guageP gh

GUAGE PRESSURE is relative to atmospheric pressure.

It is found by measuring h with a manometer.

ABSOLUTE PRESSURE is the total pressure. Pressures ADD!

Water seeks its own level – unless there is pressure!

Tower Power

How high above the ground must the

water level be in the tower to create a

gauge pressure of 3 x 105 Pa

(~ 3 atm) in your house below?

waterP gh

water

Ph

g

5 2

3 2

3 10 /

1000 / 9.8 /

x N mh

kg m x m s

30.6h m

It takes approximately

4.5 psi (.3atm) for a one-

inch diameter pipe to

push a column of water

ten feet in the air, the

standard distance

between the floors of a

house.

Zero Pressure: Making A Vacuum Mechanical Vacuum Pump

Minimum pressure produced by mechanical pump:~1Pa

Minimum pressure produced by hi tech: 10-12 Pa

Zero pressure not allowed by Quantum Uncertainty!

Absolute pressure cannot be negative: Pressure pushes not pulls!

Gauge pressure can be negative because it is a relative pressure.

Nature abhors a Vacuum.

-Aristotle

Accelerating or Decelerating?

A little girl holding a helium-filled balloon travels in a car

moving with constant velocity. The windows are up and the car

is relatively airtight. The driver then changes speed. From the

picture did the driver hit the gas or the break? Explain.

Question: Accelerating or

Decelerating?

The car accelerates. The girl’s inertia causes her to jerk back as

the car accelerates forward under her. Similarly, the air in the

car jerks back, creating a low pressure area in the front of the

car so the helium balloon is buoyed forward.

Answer:

Pascal’s Principle: Liquids & Gases A pressure applied to an enclosed fluid is transmitted

undiminished throughout the fluid: P = P everywhere.

1 2

1 2

F F

A A

How much pressure is transmitted in this hydraulic lever?

1 2

1 2

F F

A A

Pascal’s Principle: Liquids & Gases

2

2

10 9.8 /

1

kg m s

m

A = 1m2

98Pa

Buoyant Force

2 2 1 1P A PA

2 1gh A gh A

2 1gA h h

2 1

fluid

displaced

mgA h h

V fluid

displaced

displaced

mgV

V

fluidm g

The difference in pressures at different

depths produces a net upward or

Buoyant Force:

Weight of fluid displaced by object!

2 1BF F F

fluidBF W

The buoyant force on an object equals the

weight of the fluid it displaces.

Archimedes Principle

Note:

For the same volume of displaced

fluid denser fluids exert a greater

buoyant force. This is why ships float

higher in salt water than fresh water.

fluidW Vg

Archimedes Principle

The wood block is floating in the water and

has a mass of .08 kg. What is the volume of

the displaced water?

fluid displacedBF W

fluid fluid blockV g m g

/fluid block fluidV m

blockBF W

fluid displaced blockW W

3.08 /1000 /kg kg m 5 38.0 10fluidV x m

fluid objectm g m g

fluid objectm m

Since object floats, BF

(weight of fluid displaced)

equals the weight of the

object. So the masses of the

object and fluid displaced

are the same.

fluidBF W

The buoyant force on an object equals the

weight of the fluid it displaces.

Archimedes Principle

Wobject < BF Object will Float

Wobject > BF Object will Sink

Wobject = BF Object will Be Neutral

Archimedes Principle

Object will Float if:

object object fluid displacedV g V g

object displaced fluid displacedV g V g

object fluid

But the Volume of the object is the same as the

Volume of the fluid displaced:

An object will float if its density is less than the fluid it is in!

object fluid dispalcedW BF W

Archimedes Principle

object fluid

object fluid

object fluid

Sink

Float

Neutral

Archimedes Principle

Buoyant force also depends on SHAPE which

determines the VOLUME of displaced fluid.

This is how Iron Ships can float!!

They have a shape that displaces enough water to produce a

Buoyant Force to balance the weight!

Buoyant Force Question

A boulder is thrown into a deep lake. As

it sinks deeper and deeper into the water,

does the buoyant force on it increase?

NO! The buoyant force does not change as the boulder

sinks because the boulder displaces the same amount of

fluid no matter how deep it is in the water!

Question

A piece of iron placed on a block of wood makes it float

lower in the water. If the iron were instead suspended

beneath the wood, would it float as low, lower or higher?

Question A piece of iron placed on a block of wood

makes it float lower in the water. If the iron

were instead suspended beneath the wood,

would it float as low, lower or higher?

Answer

Higher. When the iron is on top, its whole weight pushes the

wood into the water. But when the iron is submerged,

buoyancy on it reduces its effective weight and less of the

block is pulled beneath the water line.

Archimedes Principle

Fraction of Object Floating

displaced

object

V

V

/

/

fluid fluid

object object

m

m

object

fluid

submerged

object

V

VSince object floats, BF

(weight of fluid displaced)

equals the weight of the

object. Thus the masses of

the object and fluid

displaced are the same.

fluid objectm g m g

fluid objectm m

Archimedes Principle

Fraction of Object Floating

submerged

object

V

V

If the above object is floating in water, what is it’s density?

object

fluid

What is the fraction submerged? 1/2

What is the density in terms of water? 1/2

Archimedes Principle

Fraction of Object Floating

submerged

object

V

V

What is the density of the object?

object

fluid

What fraction is submerged? ~1/3

Density is 1/3 that of water.

Why does the water stop when

the top is closed?

Pressure is greater in the fluid at the spout due to weight of water

so water flows. Hand covers top and water keeps flowing until

the pressure is reduced to 1 atm by increasing volume of air

above the fluid just like with a closed barometer!

How High?

Flow Rate The volume of fluid per second that flows through an area A.

VF Av

t

Continuity Equation Incompressible Fluid ( doesn’t change)

1 1 2 2Av A v

1 2v v

Continuity Equation

1 2v v

1v

2v

Fluid Flow A swelling in a blood vessel doubles its cross sectional area. By

how much will the blood flow change as it moves through the

enlarged area? The speed of blood (density = 1060 kg/m3)

through a normal part of the vessel is v = .5 m/s.

11 1 2 2 2 1

2

10.5 / 0.25 /

2

AA v A v v v m s m s

A

Continuity Equation

Calculate the velocity to which water can be squirted with the

hose with a constant flow rate of 0.500 liters per second if it :

a) emerges from the hose (r = 0.900cm) without the thumb

b) emerges with the thumb reducing the radius to r = 0.250 cm.

c)What is the maximum height the water can reach if it is shot

straight up with the thumb?

Water with air bubbles flows through a pipe

that gets narrower. In the narrow region the

water gains speed and the bubbles are

a) larger

b) smaller

c) the same size

Bernoulli’s Bubbles

Lower pressure!!

WHY?

Bernoulli’s Principle

Where the speed of a fluid increases,

internal pressure in the fluid decreases.

Bernoulli’s Ball

Why does this happen?

Bernoulli’s Principle

Bernoulli’s Equation

2

1 1 1

1constant

2Fd mv mgh

211 1

1constant

2

Fd m mv gh

V V V

2

1 1 1

1constant

2P v gh

Conservation of energy for a smooth steady fluid flow.

constantW KE PE

Efflux Speed A large tank is filled with water up to a depth of 15 m above

ground. A spout located 10.0 m above the bottom of the tank is

then opened. With what speed will water emerge from the spout?

The tank is so large that the liquid level at the top (open to air)

changes very slowly. The pressures are both at 1 atm.

2 2

1 1 1 2 2 2

1 1

2 2P v gh P v gh

1

2

2

1 2 2

1

2gh v gh

2 1 22 ( )v g h h

22 9.8 / (5 )m s m

2 9.9 /v m s

Why does the water stop when

the top is closed?

Pressure is greater in the fluid at the spout due to weight of

water so water flows. Hand covers top and water keeps

flowing until the pressure is reduced to 1 atm by increasing

volume of air above the fluid.

Bernoulli’s Principle: LIFT

Sailing Problem Calculate the approximate force on a square meter of sail, given

the horizontal velocity of the wind is 6.00 m/s parallel to its

front surface and 3.50 m/s along its back surface. 31.29 /Air kg m

( )Front BackF P P A

Question Here is a glass of ice water with an ice cube floating in it.

What fraction of the ice cube is submerged in the water?

3 3917 / , 1000 /ice waterkg m kg m

Question Here is a glass of ice water with an ice cube floating in it.

What fraction of the ice cube is submerged in the water?

submerged

object

V

V

object

fluid

3

3

917 /.917

1000 /

object

fluid

kg m

kg m

90%

3 3917 / , 1000 /ice waterkg m kg m

Question The glass is full to the brim with ice water and ice cubes.

When the ice melts, will the water level decrease? Overflow?

Remain the same? WHY?

Question Here is a glass of ice water with an ice cube floating

in it. When the ice cube melts, will the water level

rise, drop or remain the same? Assume constant

temperature.

Since the cube floats, the BF equals its weight. BF = Wice

Weight of the water displaced equals the weight of the ice cube.

mwaterg = miceg

Mass of the water displaced equals the mass of the ice cube.

When the ice melts, it is water and since it has the same mass and

density as the displaced water, it has the same volume, filling up the

space where of the displaced water. The level remains unchanged.

W

BF

Answer

Volume above and below

How does the volume of the billions of

hexagonal open spaces in the structures of ice

crystals in a piece of ice compare to the portion

of ice that floats above the water line?

water =1000 kg/m3

ice = 917 kg/m3

Answer: The volume is the same! When the ice melts, the open

spaces are filled in by the amount of ice that extends above the water

level. This is also why the water level doesn’t rise when ice melts.

If you release a ball inside a freely-falling elevator, it

stays in front of you instead of "falling to the floor"

because you, the ball, and the elevator are all

accelerating downward at the same acceleration, g. If

you similarly release a helium-filled balloon, the

balloon will

a) also stay in front of you

b) press against the ceiling

c) press against the floor

If you release a ball inside a freely-falling elevator, it

stays in front of you instead of "falling to the floor"

because you, the ball, and the elevator are all

accelerating downward at the same acceleration, g. If

you similarly release a helium-filled balloon, the

balloon will

a) also stay in front of you

b) press against the ceiling

c) press against the floor

Question

Answer: a) The balloon stays in front of you because

it, along with you and the air and the elevator, is in free fall.

When the air is in free fall it has no weight, therefore it cannot

provide a buoyant force. Therefore, the balloon falls with

everything else.