Chapter 14:
Fluids
Lecture 31
11/13/2009
Measuring Pressure: Hg Barometer
Consider what happens when we turn a fluid
filled tube upside down over a container
filled with the same fluid:
The liquid will not all fall down
Some of it will flow out of the
tube create a pocket of “vacuum”
at the top
The height of the “vacuum” pocket
is proportional to the pressure
outside the tube
... Let’s see how
Measuring Pressure: Hg Barometer
Recall from last lecture: The pressure a
distance h below the surface is given by:
p = psurface + !gh (h = depth below the
surface)
Vacuum
ptop = 0
The pressure at the level of the
atmosphere is: pbott = ptop + !gh
patm = pbott, (by definition) and
ptop = 0 (vacuum)
patm = 0 + !gh
patm = !gh
Atmosphere
pbott = patm
h
Measuring Pressure: Hg Barometer
The atmoshperic pressure is given by:
patm = gh
Vacuum
ptop = 0
This is independent of the shape/area
of the tube, only on the height of the
fluid. See fig 14-5 in the text book
Mercury (Hg) was a very commonly used
fluid in barometers. At sea level, the
height of the mercury column,
corresponding to a
Atmosphere
pressure of 1 Atmosphere pbott = patm
is 760 mm = 29.92 in Hg
h
Measuring Pressure: Open Tube Manometer
What if you want to measure the pressure
of a fluid contained in a small volume. The
previous method in not practical.
The “open tube manometer”
measures the gauge pressure
of a gas (i.e. the difference
between the gas pressure and
atmospheric pressure)
Measuring Pressure: Open Tube Manometer
What is the pressure of the gas in the tank?
a) Pressure at point 1 is just the atmospheric pressure p0:
p1 = p 0
b) The pressure at point 2 (which)
is a depth h below point 1 is:
p2 = p1 + !gh = p0 + !gh
1.
c) The pressure a point 3 (the gas
pressure) is the same as point 2
because they are at the same depth:
pgas = p3 = p2 = p0 + !gh
3.
pgauge = pgas - p0 = !gh is the “gauge”
pressure
*Note that pgauge can be +ve or -ve, h can be above/
below point 3.
2.
Gauge Pressure
You want to empty out the mud from a swamp
behind your house (to build a brand new golf
course), What gauge pressure must the
excavation machine produce to suck mud of
density 1800 kg/m3 out to a height of 1.5 m?
a) p2 = p1 + !gh
patm = psuction + !gh
b) pg ! psuction - patm
= (patm - !gh) - patm
= - !gh
= - (1800 kg/m3)(9.81m/s2)(1.5 m)
~ - 26000 kg/(ms2)
~ - 26000 N/m2
~ - 26 kPa
G
p1 = psuction
h
p2 = patm
Pascal’ Principle & The Hydraulic lever
A change in the pressure applied to an
enclosed incompressible fluid is transmitted
undiminished to every portion of the fluid
and to the walls of the container.
!pany point = !pexternal
pext
p
Pascal’ Principle & The Hydraulic lever
An important application of Pascal’s principle
is force amplification.
If you push down with a force Fi at the input
plunger (area Ai), what happens at the output
plunger (are Ao)?
!pi = !pexternal = Fi/Ai
!po = !pexternal = Fo/Ao
Fo/Ao = Fi/Ai
Fo = Fi·(Ao /Ai) > Fi
That’s how you can lift a
car to change a tire e.g.
Pascal’ Principle & The Hydraulic lever
The work done by Fi, however, is NOT amplified.
Note that the fluid is incompressible. So the volume
displaced at Vi = the volume displaced at Vo
Vi = Aidi , Vo = Aodo
Recall: W = F·d
Wi = Fi·di = (piAi)di = pi Vi
Wo = Fo·do = (poAo)do = po Vo
... but pi = po and Vi = Vo so
Wi = Wo
Wi HAS TO EQUAL Wo
otherwise you’d be creating
energy out of nothing
Buoyancy Force
: i.e. Why do some things float to the top, sink
to the bottom or stay suspended in the middle?
Archimedes principle:
When a body is partially or fully
submerged in a fluid a buoyant
force from the surrounding fluid
acts on the body.
The buoyancy force is upward* (always)
It is applied at the center of mass
The magnitude is equal to the weight
of the displaced fluid
Buoyancy Force
Calculating the Buoyant Force:
Consider a block of material the is submerged in a
fluid at a point in time. This block could be either
Accelerating down (sinking)
Accelerating up (floating)
Not accelerating (neutrally buoyant)
Fbuoy = F2 - F1 = p2A - p1A (recall last lecture)
= (p0 + !fgh2) A - (p0 + !fgh1) A
= !fgh2A - !fgh1A = !f(h2 - h1) Ag
Fbuoy = !fV g = + mfg "
Fg = -mblock g "
Fnet = Fbuoy + Fg = (mf - mblock)g "
F1
h1
h2
Fg
F2
Buoyancy Force
Calculating the Buoyant Force:
Fbuoy + mfg "
(mf = mass of the displaced fluid)
Fg = -mblock g "
Fnet = Fbuoy + Fg = (mf - mblock)g " = (!f - !block)Vg "
If:
!block < !f (the object is less dense) Fnet = +ve
it floats
!block > !f (the object is more dense) Fnet = -ve
it sinks
!block = !f (the object equally dense) Fnet = 0
it is
neutrally buoyant
F1
h1
h2
Fg
F2
Oops ...
A friends borrows your iPhone and accidentally
drops it in a pool. As you watch your beloved phone
sink you quickly calculate: (* the iPhone has dimensions of
115 mm x 62 mm x 12 mm and weighs 135 g)
a) What is the buoyant force on the iPhone
b) What is the acceleration of the iPhone towards the
bottom?
Oops
a) What is the buoyant force on the iPhone?
Determine the volume and density of the iPhone:
V = 115 mm x 62 mm x 12 mm = 85560 mm3 = 85.6 cm3
!iPhone = M/V = 135g/85.6 cm3 = 1.58 g/cm3
Fbuoy = mfg = (!waterV)g, !water = 1 g/cm3
b) What is the acceleration of the iPhone
F
towards the bottom?
F
Fy,net = +Fbuoy - Fg
= (!waterV)g - miPhoneg
= (!waterV)g - (!iPhoneV) g = (!water - !iPhone)V g
= (1 g/cm3 - 1.58 g/cm3) (85.6 cm3) (981 cm/s2)
= (- 0.58 g/cm3) (85.6 cm3) (981 cm/s2) = -48705 g cm/s2
ay = Fy,net/miPhone = (-48705 g cm/s2)/135 g = -361 cm/s2 = -3.61 m/s2
buoy
g
Floating Ice
Consider an ice cube floating in a glass of water
(or an iceberg floating in the arctic ocean).
a) What volume fraction of the ice cube is below the
water?
hi
Floating Ice
a) What volume fraction of the ice cube is below the water?
Since the ice is floating, the net force on it is zero.
Fnet = Fbuoy - Fg = mf g - mice g = (!waterVf - !iceVice)g
= (!water[hbelowA] - !ice [htotA])g = (!water hbelow - !ice htot) Ag = 0
(!water hbelow - !ice htot) Ag = 0
(!water hbelow - !ice htot) = 0
hbelow = !ice htot/!water = (!ice/!water) htot
!ice = 917 kg/m3, !water = 998 kg/m3
F1
hbelow/htot = (!ice/!water) = 917/998
= 92%
So, 92% of the ice cube is below
water and 8% is above.
habove
0
htot
Fg
hbelow
F2