Transcript Lecture 10
Liquids and Gasses Matter that “Flows”
Density and Specific Gravity
Density is mass per unit volume
m V
SI unit is
kg m
3 But this is also used
g cm
3
Specific Gravity
Ratio of density to the density of water at 4 o C.
Density of water at 4 o C=1 g/cm 3 The 4 o C is important since water’s density changes with temperature It has nothing to do with gravity Relative density would be a better term
Pressure
A static fluid will exert a force perpendicular to any surface it is in contact with Pressure is defined as Force per unit Area
p
F
A
Note this is scalar SI unit of N/m 2 =pascal=pa
Atmospheric Pressure
The big volume of air above us exerts pressure downwards.
p
Pressure at sea level 1
atm
1 .
013
x
10 5
pa a
Changes depending on elevation, temperature, weather condition etc.
Pressure at depth
constant all throughout the liquid. Fluid is incompressible Force is from weight of liquid
p p
F
A hg
mg A
Vg A
Ahg A
Pressure at depth
If fluid is compressible (gasses) density changes with height and equation is only valid for very thin slices.
Pascal’s Law
Equations independent of Area, only height matters.
Pressure is the same at any two points at the same level of the fluid.
If pressure is applied to one end it is distributed throughout the fluid and to the walls of the container
Pascal’s Law Application
Hydraulic Lift
p
F
1
A
1
F
2
A
2 However, volume of the liquid remains the same
V
1
A
1
x V
2 1
A
2
x
2
Hydraulic Lift
F
1
F
2
A
1
A
2
x
2
x
1
F
1
x
1
F
2
x
2 Work done by F1 = Work done by F2
Example
A car lift used in a service station uses compressed air to exert a force on a small piston with a circular cross section of radius 5.00cm. The pressure is transmitted by a liquid to a second piston of radius 15.0 cm. What force must the compressed air exert to lift a car weighing 13300N? What air pressure produces this force?
Example
p
F
1
A
1
F
2
A
2
F
1
A
1
A
2
F
2
r
1 2
r
1 2
F
2
F
1 5 2 13300 15 2
F
1480
N
1 Note: no need to convert to meters since units cancel
p
F
1
A
1
F
2
A
2
p
1480
N
( 0 .
05 ) 2 188000
Pa
Absolute Pressure and Gauge Pressure
Absolute Pressure Total Pressure within a vessel Gauge Pressure Excess Pressure above atmospheric pressure Gauge Pressure – Ex. A tire shows 220kPa Absolute Pressure of tire is 220kPa+1atm=320kPA Flat tire actually has 1 atm of pressure inside
Pressure Gauges - Manometer
Many different devices used to measure pressure Manometer – consists of a U-shaped tube filled with a liquid of known density. It is open in one end and the other end is connected to a container filled with gas whose pressure to be measured.
Pressure Gauges - Manometer
Bottom of tube has same pressure
p
p
p
gy
1
p atm
p atm
g
(
y
2
gy y
1 ) 2
p atm
gh
Gauge pressure is proportional to height difference
Pressure Gauges - Barometer
Mercury barometer – consists of filled tube of mercury closed on one end and the other end is inverted into a dish of mercury. Pressure in the closed end of the tube can be approximated to be zero.
Pressure Gauges - Barometer
p atm p atm
gh
gh p
0 mmHg or Torr, is another common unit of measure of pressure. But it is affected by temperature (density of Hg changes) as well as gravity.
Buoyancy
Bodies immersed in water will weigh less Bouyant Force is exerting an upward force Assume a submerged object with uniform cross section area A
Bouyancy
B
B
B
B
B
F B
F T P B A
P T A
(
gh B
A
A
(
P B
gh T
)
A
g
(
h
)
Vg B
mg
P T
)
Bouyancy
Archimedes’ Principle When a body is immersed in a fluid, the fluid exerts an upward force on the body equal to the weight of the fluid displaced by the body.
Example
Archimedes was supposedly tasked to determine if the crown of the king was pure gold. He solved this by first weighing the crown in air, and then weighing the crown when submerged in water. Suppose the scale read 7.84N in air and 6.84N in water, what is the crown made of?
Example
F
0 0
T
1
mg T
1
mg
0
T
2
B
mg T
2
mg
B T
1 7 .
84 1
T
2
B
6 .
84
w Vg
w Vg
w Vg
1
w Vg V V
1
w g
1 1000 ( 9 .
8 ) 1 .
02
x
10 4
m
3
c
m c V
7 .
84 ( 1 .
02
x
10 4 ) 9 .
8
c
gold
7840
kg m
3 19300
kg m
3
Bouyancy
B
An object will float if
W object
l V object g
m object g
l V object g
l
object
object V object g
Fluid Flow – Fluid Dynamics
Complicated, but can be simplified with certain assumptions Ideal Fluid Incompresible- same density throughout Not viscous - has no internal friction Minimal turbulence
Fluid Flow
Assume the flow is steady
flow
m flow flow
t
V
t
Av
A
l
t
Flow rate is constant 1
A
1
v
1 2
A
2
v
2
Fluid Flow
1
A
1
v
1 2
A
2
v
2 Since incompresible
A
1
v
1
A
2
v
2 Continuity Equation
Example
A water hose of 2.50cm diameter is used by a gardener to fill a 30.0L bucket. The gardener notes that it takes 1.00min to fill the bucket. A nozzle with an opening of cross-sectional area 0.500cm
2 is attached and held so water is projected horizontally from a point 1.00m above ground. Over what horizontal distance can the water be projected?
Example
Flow
A
1
v
1 30
L
0 .
03
m
min 0 .
0005
m
3
s
60
s
3
Av
0 .
0005
m
3
s v
0 .
0005
m
3
s
0 .
00005
m
2 10 .
0
m s
Example
v
10 .
0
m s t y
1 2
gt
2 2 /
g
2 /( 9 .
8 )( 1 )
t
0 .
45
s R R
v x t
4 .
5
m
10 ( 0 .
45 )
Bernoulli’s Equation
Pressure in a fluid depends on velocity and height Assuming ideal fluid, fluid at certain cross section will do work on other parts of the fluid.
W W
2 1
F
1
d
1
F
2
d
2
p
1
A
1
d
1
p
2
A
2
d
2
Bernoulli’s Equation
W g
mgh
mg
(
y
2
y
1 )
W
W
1
W
2
W g W
K
p
1
A
1
d
1
p
1
A
1
d
1
p
2
A
2
d
2
p
2
A
2
d
2
mg
(
y
2
mg
(
y
2
y
1 )
y
1 ) 1
mv
1 2 2
m
V
p
1
A
1
d
1
Ad
mgy
1 1 2
mv
2 2
p
2
A
2
d
2
mgy
2 1 2
v
1 2
p
1
gy
1 1 2
v
2 2
p
2
gy
2
Example
Water enters a house through a pipe with an inside diameter of 2.0 cm at an absolute pressure of 4.0x10
5 Pa. A 1.0cm diameter pipe leads to the second floor bathroom 5.0 m above. When the flow speed at the inlet pipe is 1.5 m/s, find the flow speed, pressure and volume flow in the bathroom.
Example
A
1
v
1
A
2
v
2
v
2
v
2
A
1
v
1
A
2 ( 0 ( 1 ) 2 .
5 ) 2 1 .
5 6
m s
1 2
v
1 2
p
1
gy
1 1 2
v
2 2
p
2
gy
2 1 2 ( 1000 )( 1 .
5 ) 2
p
2 3 .
34
x
10 5
Pa
4 .
0
x
10 5 0 1 2 ( 1000 ) 6 2
p
2 ( 1000 )( 9 .
8 )( 5 )
Example
Flow Flow
A
1
v
1 ( 0 .
5
x
10 2 ) 2 6
Flow
4 .
7
x
10 4
m
3
s
Problems
Giancoli 10-20 In working out Pascal’s principle, Pascal showed dramatically how force can be multiplied with fluid pressure. He placed a long thin tube of radius r=0.30cm, vertically into a wine barrel of radius R=21cm. He found that when the barrel was filled with water and the tube filled to a height of 12m, the barrel burst. Calculate (a) the mass of water in the tube? (b) the net force exerted on the barrel lid just before rupture.
Serway 14-20
A U tube of uniform cross-secitonal area, open to the athmosphere, is partially filled with mercury. Water is then poured into both arms. If the equilibrium configuration fo the tube is as shown with h 2 =1.00cm, find h 1 .
Young and Friedman 14.31
A cubical block of wood, 10 cm on a side, floats in the interface of oil and water with its lower surface, 1.50cm below the interface. The density of oil is 790kg/m 3 . (a) what is the gauge pressure at the upper face of the block? (b) What is the gauge pressure at the lower face of the block? (c) What are the mass and density of the block?
Young and Friedman 14.46
A golf course sprinkler system discharges water from a horizontal pipe at the rate of 7200cm 3 /s. At one point in the pipe, where the radius is 4.00cm, the absolute pressure is 2.40x10
5 Pa. At a second point in the pipe, the water passes through a connection where the radius is 2.00cm. What is the absolute pressure as it flows through the constriction?