Lecture 2Fluids: density, pressure,

Pascal’s principle.

What is a fluid?

Fluids are “substances that flow”…. “substances that take the shape of the container”Fluids are “substances that flow”…. “substances that take the shape of the container”

Atoms and molecules must be free to move .. No long range correlation between positions (e.g., not a crystal).

Atoms and molecules must be free to move .. No long range correlation between positions (e.g., not a crystal).

Gas or liquid… or granular materials (like sand)Gas or liquid… or granular materials (like sand)

Density, pressure

Vm

Density:

Fp

APressure:

Units: Pascal (Pa) = 1 N/m2

psi (pounds per square inch)

atmosphere 1 atm = 1.013 × 105 Pa

bar 1 bar = 105 Pa

Units: Pascal (Pa) = 1 N/m2

psi (pounds per square inch)

atmosphere 1 atm = 1.013 × 105 Pa

bar 1 bar = 105 Pa

Pure water: 1000 kg/m3Pure water: 1000 kg/m3

Atmospheric pressure

DEMO: Piston and weight

The atmosphere of Earth is a fluid, so every object in air is subject to some pressure.The atmosphere of Earth is a fluid, so every object in air is subject to some pressure.

At the surface of the Earth, the pressure is

patm ~ 1.013 x 105 Pa = 1 atm

At the surface of the Earth, the pressure is

patm ~ 1.013 x 105 Pa = 1 atm

Area of a hand ~ 200 cm2 = 0.02 m2

Area of a hand ~ 200 cm2 = 0.02 m2

atm ~2000 N on your hand due to air!F p A

Pressure vs. depth

Fbottom

Ftop

mg

Imaginary box of fluid with bases of area A

and height h

Imaginary box of fluid with bases of area A

and height h

bottom/ top

bottom/ top

FP

A

bottom topp p gh

m Ah

hNet force must be zero!Net force must be zero!

bottom topF F mg

DEMO: Plastic tube with cover

5

bottom top

3 3 2

Example: How deep under water is = 2 atm?

1.01 10 Pa10.3 m

10 kg/ m 9.81 m/ s

(ie, 1 atm is produced by a 10.3 m high column of water)

p

p ph

g

Fluid in an open container

Pressure is the same at a given depth, independently of the container. p(y)

y

Fluid level is the same everywhere in a connected container (assuming no surface forces) •

A•B

If liquid height was higher

above A than above B

If liquid height was higher

above A than above B

pA > pB pA > pB Net force

Net force

Net flow

Net flow

This is not equilibriu

m!

This is not equilibriu

m!

DEMO: Pascal’s

vases

ACT: U tube

Two liquids Y and G separated by a thin, light piston (so they cannot mix) are placed in a U-shaped container. What can you say about their densities?

Two liquids Y and G separated by a thin, light piston (so they cannot mix) are placed in a U-shaped container. What can you say about their densities?

A. ρG < ρY

B. ρG = ρY

C. ρG > ρY

A. ρG < ρY

B. ρG = ρY

C. ρG > ρY

YY

GG

•A

•B

Pressure at A and B must be the same:Pressure at A and B must be the same: Y 1 G 2 atm G 3 atmgh gh p gh p

h1

h2

h3

Y 1 G 3 2h h h 1 3 2 Y GSince h h h

DEMO: U-tube with

water and kerosene

Water towers

Water towers are a common sight in the Midwest… because it’s so flat!

Water towers are a common sight in the Midwest… because it’s so flat!

hh

house atm waterp p hg

So physics sucks, but how much?

Your physics professor sucks on a long tube that rises out of a bucket of water. She can get the liquid to rise 5.5 m (vertically). What is the pressure in her mouth at this moment?

Your physics professor sucks on a long tube that rises out of a bucket of water. She can get the liquid to rise 5.5 m (vertically). What is the pressure in her mouth at this moment?

A. 1 atm

B. 0.67 atm

C. 0.57 atm

D. 0.46 atm

E. 0 atm

A. 1 atm

B. 0.67 atm

C. 0.57 atm

D. 0.46 atm

E. 0 atm

DEMO: Sucking

through a hose

water atmmouth

atm watermouth

5 3 3 2 10 Pa 10 kg/ m 9.8 m/ s 5.5 m

46100 Pa 0.46 atm

p gh p

p p gh

hh

x Ax A

x Bx B

Pascal’s principle

Any change in the pressure applied to an enclosed fluid is transmitted to every portion of the fluid and

to the walls of the containing vessel.

Pascal’s Principle is most often applied to incompressible fluids (liquids):

Increasing p at any depth (including the surface) gives the same increase in p at any other depth

Pascal’s Principle is most often applied to incompressible fluids (liquids):

Increasing p at any depth (including the surface) gives the same increase in p at any other depth

Hydraulic chamber

F F

1

2

d

2d

1

A A21

1 2

1 2

F F

A A

22 1

1

AF F

A F2 can be

very large…F2 can be

very large…

1 1W Fd

No energy is lost:No energy is lost:

1 1 2 2I ncompressible fl uid: Ad A d

1 22 2 2 2

2 1

A AF d F d

A A

ACT: Hydraulic chambers

In each case, a block of mass M is placed on the piston of the large cylinder, resulting in a difference di between the liquid levels. If A2 = 2A1, then:

In each case, a block of mass M is placed on the piston of the large cylinder, resulting in a difference di between the liquid levels. If A2 = 2A1, then:

A1 A10

A2 A10

M

MdB

dA

A. dA < dB

B. dA = dB

C. dA > dB

A. dA < dB

B. dA = dB

C. dA > dB

Measuring pressure with fluids

Barometer Measures absolute pressure Top of tube evacuated (p = 0) Bottom of tube submerged into pool of

mercury open to sample (p) Pressure dependence on depth:

Hg

ph

g

vacuum

p=0

h atmosphere

p=p0

Barometer

Sample at p

Sample at p

hh

Vacuum p =

0

Vacuum p =

0

pp patmpatm

∆h∆h

p0

h

Manometerp1

Manometer Measures gauge pressure: pressure relative to

atmospheric pressure. Pressure dependence on depth:

atm

Hg

p ph

g

A unit for pressure

760 mm Hg = 1 torr = 1 atm