Chapter 10
Fluids
Pascal’s Principle
Example: A Car Lift -- revisited
The input piston has a radius of 0.0120 m
and the output plunger has a radius of
0.150 m.
The combined weight of the car and the
plunger is 20500 N. Suppose that the input
piston has a negligible weight and the bottom
surfaces of the piston and plunger are at
levels so that h = 1.10 m in the figure.
Also assume that hydraulic oil is used in the
car lift with density ρ = 800 kg/m3.
What is the required input force of the piston
to balance the weight of the car + plunger
in this configuration?
A2
Pascal’s Principle
F1 = P1 A1
P2 = P1 + ρ gh ⇒ P1 = P2 − ρ gh
F2 A1
∴ F1 = ( P2 − ρ gh ) A1 =
− ρ ghA1
A2
=
2
20500
π
0.0120
(
) (
)
π ( 0.150 2 )
− (800 ) ( 9.80 ) (1.10 ) π ( 0.0120 2 )
= 131 N − 4 N = 127 N
A2
Pressure Gauges
A mercury barometer – used to
measure atmospheric pressure
P2 = P1 + ρ gh
Patm = ρ gh
Patm
1.01×10 5 Pa
h=
=
ρ Hg g (13.6 ×10 3 kg/m 3 ) ( 9.80 m/s2 )
= 0.760 m = 760 mm = 29.92 inches
If water were used instead,
Patm
1.01×10 5 Pa
h=
=
ρ water g (1000 kg/m 3 ) ( 9.80 m/s2 )
= 10.3 m A much larger column!
Pressure Gauges
P2 = PB = PA
PA = P1 + ρ gh
absolute pressure
P2 − Patm = ρ gh
$
!#!
"
gauge pressure
Open-tube manometer – used to measure
the pressure, P2, in a container
Pressure Gauges
Stickies
Measuring blood pressure
with an open-tube manometer
Typical blood pressures:
peak (systolic): 120 mm Hg
rest (diastolic): 80 mm Hg
Pressure Gauges
Example: A Tire Pressure Gauge
The spring constant of the spring
is 320 N/m and the bar indicator
extends 2.0 cm. What force does the
air in the tire apply to the spring?
What pressure in psi does this reading
correspond to if the diameter of the
air chamber cross section is 6.38 mm?
Pressure Gauges
FxApplied = k x
= (320 N m ) ( 0.020 m ) = 6.4 N
FxApplied
6.4
6.4
5
P=
=
=
=
2.00
×10
Pa
2
−5
A
! 0.00638 $ 3.20 ×10
π#
&
" 2 %
Since 1 pound-per-square-inch (psi) ≅ 6895 Pa
1 psi
∴ P = 2.00 ×10 5 Pa ×
= 29.0 psi
6895 Pa
⇒ normal tire pressure
Pressure Gauges
Aneroid barometers
Weather barometer
Reads out in millibars
or inches of Hg
(1 atm = 1013 millibars)
The aneroid cell, a spring-metal bellows evacuated of air,
expands or contracts depending on the atmospheric pressure.
Aviation altimeter
Reads out in thousands
of feet above sea-level
sea−level
h
Patm
≈ Patm
+ ρ air gh
sea−level
h
Patm
− Patm
h≈
ρ air g
Archimedes’ Principle
Consider an object, e.g. a cylinder, totally immersed
in a fluid à calculate the net force on it by the fluid:
P2 − P1 = ρ gh
Buoyant force
FB = P2 A − P1 A = (P2 − P1 )A
V = hA
FB = ρ ghA
FB = !
ρV g
mass of
displaced
fluid
Archimedes’ Principle
ARCHIMEDES’ PRINCIPLE
Any fluid applies a buoyant force to an object that is partially
or completely immersed in it; the magnitude of the buoyant
force equals the weight of the fluid that the object displaces:
F
!B
Magnitude of
buoyant force
= Wfluid
!
Weight of
displaced fluid
Archimedes’ Principle
If the object is floating then the
magnitude of the buoyant force
is equal to the magnitude of its
weight.
Not floating:
FB < mobject g
Floating:
FB = mobject g
Minimum requirement for floating:
FB,max = ρ fluidVobject g = mobject g = ρobjectVobject g
⇒ ρ fluid = ρobject
⇒ Object floats if ρ fluid ≥ ρobject
Archimedes’ Principle
Example: A Swimming Raft
The raft is made of a solid square
of pinewood. Determine whether
the raft floats in water and if so,
how much of the raft is beneath
the surface.
Archimedes’ Principle
Vraft = ( 4.0 m ) ( 4.0 m ) ( 0.30 m ) = 4.8 m 3
FBmax = ρ Vg = ρ waterVwater g
(
)(
)(
= 1000 kg m 3 4.8m 3 9.80 m s 2
= 47000 N
)
Archimedes’ Principle
Wraft = mraft g = ρ pineVraft g
(
)(
)(
= 550 kg m 3 4.8m 3 9.80 m s 2
= 26000 N < 47000 N
The raft floats!
)
Archimedes’ Principle
If the raft is floating:
Wraft = FB
26000 N = ρ waterVwater g
(
)
(
26000 N = 1000 kg m3 (4.0 m)(4.0 m)h 9.80 m s 2
h=
)
26000 N
= 0.17 m
3
2
1000 kg m (4.0 m )(4.0 m ) 9.80 m s
(
)
(
)
Archimedes’ Principle
Conceptual Example: How Much Water is Needed
to Float a Ship?
A ship floating in the ocean is a familiar sight. But is all
that water really necessary? Can an ocean vessel float
in the amount of water that a swimming pool contains?
Archimedes’ Principle
A state-of-charge indicator for a
car battery.
ρcharged acid > ρball
ρdischarged acid < ρball