July 14 Th Lecture 3

8/18/2019 July 14 Th Lecture 3

1/19

Today (Chapter 10, Fluids)

Review for Exam 2

Tomorrow (Chapters 610)

Review Co!"epts from Tuesday

Co!ti!uity E#uatio!

$er!oulli%s E#uatio!

&ppli"atio!s'Examples


2/19

When an object is submerged in a fuid (air, water,etc.) it will eel a pressure rom the fuid. Thispressure is caused by the collision o molecules othe fuid with the surace o the object.

From our own experience, we now that thepressure deeper in a fuid is more than thepressure at the surace.

What is the di!erence in the pressure on the top othe object and on the bottom o the object"

P2 = P

1 + ρ gh

ressure


3/19

Quiz Question!

# submarine is constructed so that it

can saely withstand a pressure o$.% x $&' a. ow deep can thissubmarine descend in the ocean i

the a*erage density o sea water is$&+ g-m"


4/19

Fluids and /!ect o 0ra*ity

• 12Tube /xample $&. on age $%.

• 3onsider a 12Tube with both ends open to

the atmosphere. 4et there be two li5uids,water with density $&&& g-m and an oilwith density '&& g-m. 4et the oil be6lled &.$& m abo*e the oil2water boundary.

• 7etermine the di!erence in the le*el o thetwo li5uids, d.


5/19

as"al%s ri!"iple(why hydraulics wor)

#ny change in the pressure applied to acompletely enclosed fuid is transmitted

undiminished to all parts o the fuid and the

enclosing walls.

8n other words, pressure applied at one endo a hydraulic system is added to e*ery

point in the system.F$ F+#$ #+


6/19

9uoyancy(why things foat)

#ny fuid applies a buoyant orce to an object thatis partially or completely immersed in it.

The magnitude o the buoyant orce e5uals

the weight o the fuid that the object displaces.

8n other words, i the object can displace enoughfuid,

it will generate enough buoyant orce tocounteract

it:s weight and it will foat;F B =W fluid − displaced

&r"himedes% ri!"iple

gV F B ρ =


7/19

# solid bloc o wood thatnormally foats in water ispushed down and held underwater by a physics +$< student.

7oes the water le*el in thecontainer rise or all" (rise)

8s the buoyant orce on the woodgreater than, e5ual to, or lessthan the weight o the object"(greater)

9eore

#ter


8/19

# fat2bottomed barge, loaded with coal, has a mass o

=$& g. The barge is +&.& m long and $&.& m wide.8t foats in the resh water. What is the depth o thebarge below the waterline"

displaced water barge

water displaced water barge

water barge

barge

3

water

3000001.5

1000 / 20 10

B F mg

m g m g

V m

wlh m

m kg h m

wl kg m m m

ρ

ρ

ρ

=

=

=

=

= = =


9/19

# bloc o birch wood foats in oil with >&? o its

*olume submerged. What is the density o the oil" Thedensity o the birch is &.%'g-cm.

3

0.90.9

0.744 /

B birch

displaced oil birch

oil displaced oil birch birch

oil displaced oil birch birch

oil birch birch birch

oil birch

oil

F m g

m g m g

V g V g

V V

V V

g cm

ρ ρ

ρ ρ

ρ ρ

ρ ρ

ρ

=

=

=

=

==

=


10/19

What comes in, must go out.

1 1 x v t = ∆ 2 2 x v t = ∆

1 1 1 1 1 1 1m A x A v t ρ ρ ∆ = = ∆ 2 2 2 2 2 2 2m A x A v t ρ ρ ∆ = = ∆

1 1 1 2 2 2 A v t A v t ρ ρ ∆ = ∆

1 1 2 2 A v A v=1 1 1 2 2 2 A v A v ρ ρ =

3onstan

t density

Co!ti!uity E#uatio!


11/19


@ass fow rate

m Av

t ρ

∆=

∆1nitsA g-s

Bolume fow rate V

Av

t

∆=

∆

1nitsA m-s

Co!ti!uity E#uatio!

1 1 2 2 A v A v=


12/19

(why a 6re hose wors)

The mass fow rate has the same *alue at e*eryposition along a tube that has one entry point

and one exit point.

8n other words, i you reduce the area that thefuid can fow through, it has to fow aster;


(8 you sho*e + gallons o water in one end o a

pipe in one second, + gallons o water must come

ρ 1 A

1v1 = ρ

2 A

2v

2

Co!ti!uity E#uatio!

1 1 2 2 A v A v=


13/19

• 3ontinuity /5uation /xample

Water is fowing through a pipewith a cross2sectional area o C.&

cm+ and connects to a aucetwith an opening o area &.& cm+.

8 the water is fowing at aspeed o .& m-s in the pipe,what is the speed as it lea*es theaucet"

1 1 2 2

A v A v=


14/19

(why a hea*y airplane can fy)

8n a (steady, irrotational) fow o a (non*iscous,incompressible) fuid o density ρ, the pressure,fuid speed, and ele*ation (y) at any two points

are related byA

8n other words, i we are taling about two pointswith the same ele*ationA

a 5uicly fowing fuid has a lower pressurethan a slowly fowing fuid.

P1 +

1

2 ρ v

1

2

+ ρ gy1 = P

2 +

1

2 ρ v

2

2

+ ρ gy2

$er!oulli%s E#uatio!


15/19

#pplicationA 9ernoulli:s /5uation

• What is the upward DlitE orce on anairplane wing"

• #ssume the area o the wing is %& m+

• The speed o air o*er the wing is +&m-s

•

The speed o air under the wing is +&&m-s

• 4et the wing be $ meter thic

• #ssume the density o the air to be $.+>

g-m

P1 +

1

2 ρ v

1

2

+ ρ gy1 = P

2 +

1

2 ρ v

2

2

+ ρ gy2


16/19

7i!erence in ressures

• The pressure in the atmosphere is not aconstant, but fuctuates as the weatherchanges.

• 3onsider a window o a house where thepressure inside is $&$,&&& a and thepressure outside has decreased to >%&&& a.4et the area o the window be $. m+.

• What is the orce exerted on the window dueto this di!erence in pressures"


17/19

Fluids and 0ra*ity

# new water tower is built on a *ery tallhill. #t a nearby house, the pipes areleaing because the pressure in the

water is too high.

#ssume water pipes lea when thepressure exceeds 6*e timesatmospheric pressure.

7etermine the relati*e height o the

water tower in comparison to the


18/19

#rchimedes:s rinciple and

9uoyancy

• 3onsider a wooden bloc odensity '&& g-m. and a

*olume o +.& m.

.

• 7etermine the orce re5uired

to hold it completely underwater.


19/19

#rchimedes:s rinciple and

9uoyancy

• 3onceptA #pparent Weight

• 3onsider a piece o aluminum attached

to a string and suspended in a pool ooil (density '& g-m).

• 4et the apparent weight o the

aluminum be C& .

• 7etermine the *olume o the aluminum.

July 14 Th Lecture 3

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