Lecture 02 density, pressure and pascal's principle

1. Lecture 2
Fluids: density, pressure,
Pascal’s principle.

2. What is a fluid?
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).
Gas or liquid… or granular materials (like sand)

3. Density, pressure
Density:
Pressure:
Units:
m
ρ=
V
Pure water: 1000 kg/m3
F⊥
p=
A
Pascal (Pa) = 1 N/m2
psi (pounds per square inch)
atmosphere
bar
1 atm = 1.013 × 10 5 Pa
1 bar = 105 Pa

4. Atmospheric 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
Area of a hand ~ 200 cm2 = 0.02 m2
F = patmA ~ 2000 N
on your hand due to air!
DEMO: Piston
and weight

5. DEMO:
Plastic tube
with cover
Pressure vs. depth
Imaginary box of fluid
with bases of area A
and height h
Ftop
Net force must be zero!
Fbottom = Ftop + mg
Pbottom/top =
m = ρAh
h
mg
Fbottom
Fbottom/top
A
pbottom = ptop + ρ gh
Example: How deep under water is p = 2 atm?
pbottom − ptop
1.01 × 105 Pa
h=
=
= 10.3 m
3
3
2
ρg
10 kg/m 9.81 m/s
(
)(
)
(ie, 1 atm is produced by a 10.3 m high column of water)

6. DEMO:
ascal’s vases
Fluid in an open container
Pressure is the same at a
given depth, independently of
the container.
y
Fluid level is the same everywhere in
a connected container (assuming no
surface forces)
If liquid height
was higher above
A than above B
pA > pB
Net force
→
p(y)
•
A
Net flow
→
•
B
This is not
equilibrium!

7. DEMO:
U-tube with
water and
kerosene
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?
A. ρG < ρY
h1
Y
G
h2
•
A
B. ρG = ρY
h3
•
B
C. ρG > ρY
Pressure at A and B must be the same:
ρ Y gh1 + ρG gh2 + patm = ρG gh3 + patm
ρ Yh1 = ρ G ( h3 − h2 )
Since h1 < h3 − h2
⇒
ρ Y > ρG

8. Water towers
Water towers are a common sight in the Midwest… because it’s so flat!
h
phouse = patm + ρwaterhg

9. 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?
xB
h
A. 1 atm
B. 0.67 atm
x A
C. 0.57 atm
D. 0.46 atm
E. 0 atm
DEMO:
Sucking
through a
hose
pmouth + ρwater gh = patm
pmouth = patm − ρwater gh
= 105 Pa − ( 103 kg/m3 ) ( 9.8 m/s2 ) ( 5.5 m )
= 46100 Pa = 0.46 atm

10. 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

11. Hydraulic chamber
F1
F1 F2
=
A1 A2
A2
F2 = F1
A1
d2
F2 can be
very large…
d1
A1
No energy is lost:
 A1
W = F1d1 =  F2
 A

2
 A2
÷ d2
÷ A

1
F2

÷ = F2d2
÷

Incompressible fluid: Ad1 = A2d2
1
A2

12. 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:
A. dA < dB
dA
A1
M
A10
B. dA = dB
C. dA > dB
dB
A2
M
A10

13. Measuring pressure with fluids
Barometer
vacuum
Vacuum
p p=0
=0
Measures absolute pressure
Barometer
Top of tube evacuated (p = 0)
atmosphere
Bottom of tube submerged into pool of mercury
Sample
p=p
at p0
open to sample (p)
p
Pressure dependence on depth: h =
g ρHg
Manometer
Measures gauge pressure: pressure relative to
atmospheric pressure.
p − patm
∆h =
Pressure dependence on depth:
g ρHg
A unit for pressure
760 mm Hg = 1 torr = 1 atm
h
h
Manometer
p1
p
p atm
p0atm
∆h
∆h