PHY 113 A Fall 2012 -- Lecture 26 111/05/2012

PHY 113 A General Physics I9-9:50 AM MWF Olin 101

Plan for Lecture 26:

Chapter 14: The physics of fluids

1. Density and pressure

2. Variation of pressure with height

3. Buoyant forces

PHY 113 A Fall 2012 -- Lecture 26 211/05/2012

PHY 113 A Fall 2012 -- Lecture 26 311/05/2012

The physics of fluids.

•Fluids include liquids (usually “incompressible) and gases (highly “compressible”).

•Fluids obey Newton’s equations of motion, but because they move within their containers, the application of Newton’s laws to fluids introduces some new forms.

Pressure: P=force/area 1 (N/m2) = 1 Pascal

Density: r =mass/volume 1 kg/m3 = 0.001 gm/ml

Note: In this chapter Ppressure (NOT MOMENTUM)

PHY 113 A Fall 2012 -- Lecture 26 411/05/2012

Pressure

AP

F

Note: since P exerted by a fluid acts in all directions, it is a scalar parameter

PHY 113 A Fall 2012 -- Lecture 26 511/05/2012

Example of pressure calculation High heels (http

://www.flickr.com/photos/moffe6/3771468287/lightbox/)

F

Pam

N

A

mg

AP

heel

62

105.101.001.0

4/6004/

F

PHY 113 A Fall 2012 -- Lecture 26 611/05/2012

Pressure exerted by air at sea-level

1 atm = 1.013x105 Pa

Example: What is the force exerted by 1 atm of air pressure on a circular area of radius 0.08m?

F = PA = 1.013x105 Pa x p(0.08m)2

= 2040 N

Patm

PHY 113 A Fall 2012 -- Lecture 26 711/05/2012

Density = Mass/Volume

PHY 113 A Fall 2012 -- Lecture 26 811/05/2012

Relationship between density and pressure in a fluid

Effects of the weight of a fluid:

gdydP

dydP

yyPyyP

ygyyPyPAmg

AyyF

AyF

mgyyFyF

y

ρ

)()(lim

ρ)()(

)()(

)()(

0

yrgDy = mg/A

P(y+Dy)

P(y)

Note: In this formulation +y is defined to be in the up direction.

PHY 113 A Fall 2012 -- Lecture 26 911/05/2012

For an “incompressible” fluid (such as mercury):

r = 13.585 x 103 kg/m3 (constant)

)(ρ ρ 00 yygPPgdydP

= 13.595 r x 103 kg/m3

Example:

m 76.0

m/s 8.9kg/m 1013.595

Pa 10013.1

ρ

233

5

00

g

Pyyh

PHY 113 A Fall 2012 -- Lecture 26 1011/05/2012

Barometric pressure readingsHistorically, pressure was measured in terms of inches of mercury in a barometer

)(ρ ρ 00 yygPPgdydP

= 13.595 r x 103 kg/m3

in93.290.0254m

1in0.76mm 76.0

m/s 8.9kg/m 1013.595

Pa 10013.1

ρ

233

5

00

g

Pyyh

PHY 113 A Fall 2012 -- Lecture 26 1111/05/2012

Weather report:

m763.0in

0.0254min03.30in03.30

PHY 113 A Fall 2012 -- Lecture 26 1211/05/2012

Question: Consider the same setup, but replace fluid with water (r = 1000 kg/m3). What is h?

= 1000r kg/m3

iclicker equation:Will water barometer have h:

A. Greater than mercury.B. Smaller than mercury.C. The same as mercury.

PHY 113 A Fall 2012 -- Lecture 26 1311/05/2012

Question: Consider the same setup, but replace fluid with water (r = 1000 kg/m3). What is h?

)(ρ ρ 00 yygPPgdydP

= 1000r kg/m3

m34.10

m/s 8.9kg/m 0001Pa 10013.1

ρ

23

5

00

gP

yyh

PHY 113 A Fall 2012 -- Lecture 26 1411/05/2012

m 5.0

kg/m 1000 3

h

Pa 10013.1 5atmP

iclicker question:A 0.5 m cylinder of water is inverted over a piece of paper. What will happen

A. The water will flow out of the cylinder and make a mess.

B. Air pressure will hold the water in the cylinder.

PHY 113 A Fall 2012 -- Lecture 26 1511/05/2012

ρ

ρ

ρ :gas idealan For 0

0

0

0

P

gP

dy

dP

PP

mi

yy

m

yyyyP

g

ePePePyP 50

800000

0000

0

)( :Solution

)(ρ (constant) :etc mercury, For water, 00 yygPP

General relationship between P and :r

gdy

dPρ :surface sEarth'near fluids allFor

PHY 113 A Fall 2012 -- Lecture 26 1611/05/2012P (atm)

y-y 0

(mi)

mi

yy

m

yyyyP

g

ePePePyP 50

800000

0000

0

)( :Solution

Approximate relation of pressure to height above sea-level

PHY 113 A Fall 2012 -- Lecture 26 1711/05/2012

iclicker question:Have you personally experienced the effects of atmospheric pressure variations?

A. By flying in an airplaneB. By visiting a high-altitude location (such as

Denver, CO etc.)C. By visiting a low-altitude location (such as

Death Valley, CA etc.)D. All of the above.E. None of the above.

PHY 113 A Fall 2012 -- Lecture 26 1811/05/2012

Buoyant forces in fluids(For simplicity we will assume that the fluid is incompressible.)

Image from the web of a floating iceberg.

Image from the web of a glass of ice water

PHY 113 A Fall 2012 -- Lecture 26 1911/05/2012

Buoyant force for fluid acting on a solid:

FB=rfluidVdisplacedg

submergedB

topbottomB

gVyAgAyyPyPF

FFF

ygyyPyP

fluidfluid

fluid

ρρ)()(

:forceBuoyant

ρ)()(

mg

FB - mg = 0

rfluidVsubmergedg - rsolidVsolidg = 0

fluid

solid

solid

submerged

V

V

ρ

ρ

A

Dy

PHY 113 A Fall 2012 -- Lecture 26 2011/05/2012

Summary:

submergedB gVF fluidρ :forceBuoyant

fluid

solid

solid

submerged

V

V

ρ

ρ

Some densities: ice r = 917 kg/m3

fresh water r = 1000 kg/m3

salt water r = 1024 kg/m3

PHY 113 A Fall 2012 -- Lecture 26 2111/05/2012

iclicker question:Suppose you have a boat which floats in a fresh water lake,

with 50% of it submerged below the water. If you float the same boat in salt water, which of the following would be true?

A. More than 50% of the boat will be below the salt water. B. Less than 50% of the boat will be below the salt water. C. The submersion fraction depends upon the boat's total

mass and volume. D. The submersion fraction depends upon the barometric

pressure.

PHY 113 A Fall 2012 -- Lecture 26 2211/05/2012

Archimede’s method of finding the density of the King’s “gold” crown

Wwater Wair

waterair

airwaterobject

objectobjectair

objectwaterobjectobjectBwater

WWW

gVmgW

gVgVFmgW

PHY 113 A Fall 2012 -- Lecture 26 2311/05/2012

(water) 1000kg/mρ :example

)(ρ :fluid ibleIncompress

ρ

3

00

yygPP

gdydP

Application of Newton’s second law to fluid (near Earth’s surface)

(air) kg/m29.1ρ :example

)1)(ρ

(for )(ρ

:fluid leCompressib

3

00

0000

)(ρ

0

00

0

yyP

gyygP

ePPyy

P

g

Summary:

PHY 113 A Fall 2012 -- Lecture 26 2411/05/2012

iclicker question:

Suppose that a caterer packed some food in an air tight container with a flexible top at sea-level. This food was loaded on to an airplane with a cruising altitude of ~6 mi above the earth’s surface. Assuming that the airplane cabin is imperfectly pressurized, what do you expect the container to look like during the flight?

(A) (B) (C)

PHY 113 A Fall 2012 -- Lecture 26 2511/05/2012

Example:

Hydraulic press

incompressible fluid

A1Dx1=A2Dx2

F1/A1= F2/A2

PHY 113 A Fall 2012 -- Lecture 26 2611/05/2012

PHY 113 A Fall 2012 -- Lecture 26 2711/05/2012