Download Physics 1010: The Physics of Everyday Life

Physics 1010:
The Physics of Everyday Life
TODAY
• Pressure, Boyancy
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Today’s topics
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Pressure, density, and temperature
Ideal gas law
Archimedes’ principle & buoyancy force
Water pressure
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Gases
• Consist of molecules (about 10-9 m across) that
bounce around
• They do not fall to ground because they have
kinetic energy [(1/2)mv2 ]
• More kinetic energy means more temperature
• Bouncing off the wall causes it to exert a force
(concervation of momentum; Newton’s III)
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Pressure: bouncing off the walls
The piston is free to
move. After the tiny
molecule hits it, the
piston
a) shakes
b) moves to the left at
constant velocity
c) accelerates to the left
d) remains stationary
The piston moves to the left at
constant velocity, as it acquired
momentum from the molecule
(concervation of momentum).
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Pressure: bouncing off the walls
The piston is free to move.
Molecule after molecule
hits it, the piston
a) shakes
b) moves to the left at
constant velocity
c) accelerates to the left
d) remains stationary
The piston accelerates to the left as
each molecule gives it more
momentum, meaning…..
THERE IS A NET FORCE ON THE PISTON
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Pressure: the force per unit area
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More area exposed to gas implies more collisions
More collisions implies more momentum transfer
More momentum transfer implies more force
Force = Pressure x Area
Pressure = Force/Area
Unit: 1 Pascal (Pa) = 1 N/m2
Atmospheric pressure = 100,000 Pa = 105 Pa =
14.7 lbs/sq. inch
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Pressure is in all directions
• Water comes out
whether cork is on
top, side, or bottom
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Mass density: mass per unit volume
• Allows calculation of mass from volume
• Mass = Density x Volume
• Some common densities:
Water (humans and animals): 1000 kg/m3 (64 lbs/cubic
foot)
Lead: 11,400 kg/m3
Uranium: 19,000 kg/m3
Air: 1.239 kg/m3
• Squeezing air: more air in smaller volume, so
denser, more density
Mass density is denoted by ρ
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Anything can form a density
• Particle density: # of particles / unit volume
• People density: # of people /unit volume
• What is the people density in this room? (room is
~20mx15mx6m, class is ~200 students)
a) 0.05 people/m3
b) 0.1 people/m3
c) 0.5 people/m3
d) 1 people/m3
People density = number of people/volume of room
200 people/1,800 m3 = 0.11 people/ m3
Mass density is denoted by ρ, number density by n
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Pressure increases with density
• Twice the molecules
means twice the collisions
• Twice the collisions means
twice the momentum
transfer
• Twice the momentum
transfer means twice the
pressure
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Pressure increases with average kinetic
energy [(1/2)mv2]
• More speed means more
collisions with the wall
• More speed and mass, more
momentum that each collision
transfers to wall
• Pressure proportional to v x
mv ~ mv2
• Pressure proportional to
temperature
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Ideal gas law
• Pressure proportional to density x
temperature (relative to absolute zero)
Density: more dense means more molecules
hitting sides of wall
Temperature: more temperature means
molecules hitting more rapidly, with more
momentum transfer
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Ideal gas law
“is proportional to”
p∝nT
absolute pressure
(not pressure relative
to atmosphere)
absolute temperature
(use Kelvin, not F or C!)
molecule density
(number of molecules per unit volume)
Constant of proportionality: “Boltzmann’s constant”
k = 1.38 x 10-23 J/K = 1.38 x 10-23 N m/K = 1.38 x 10-23 Pa m3/ K
p=nkT
Amazingly, k is a “universal” constant:
it is the same for any species of gas.
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Can you answer?
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In higher temperature gases, the molecules move
A) faster or B) slower?
Does pressure push down as well as up?
A)True B) False
How does pressure relate to force?
A) F=mass x acceleration B) F=Pressure x Area
What is a Pascal?
What does mass density mean? Or any kind of density?
How does pressure change with mass density and
temperature?
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Properties of the earth’s atmosphere
• Earth radius = 4,000 mi
• Atmospheric thickness = 4 mi
• Less than one pixel on screen
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Atmospheric pressure is due to weight of air
• Force down on
separating membrane
is weight of air above
• Manifestation is
pressure
• Force up from air
below is pressure to
balance
• Pressure is weight of
column of air divided
by area
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Pressure under water increases more
rapidly as you go down
• Pressure is the weight of water
above one square meter
• Weight increases as you go down
• For each meter down, adding 1
cubic meter, or 1000 kg = 10,000
N
• Each 10 m then add 100,000 N or
one atmosphere!
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Buoyancy: how ballons lift and ships float
• Archimedes principle: an
object in a fluid feels an
upward force equal to the
weight of the fluid it
displaces
• Fluid falls down where
object was
• Net weight = weight of
object in vacuum minus
weight of fluid having
volume equal to that of
object
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100,000 Pa
=
Empty Cube
(with side 1 meter)
Not even air inside
m
1m
1m
1m
At sea level, atmospheric pressure is about 100,000 Pa. How much
force does the atmosphere exert on the top side of an empty cube with
side 1 meter? What mass, set on top of the cube, would exert
approximately the same force? Hint: 100,000 Pa = 100,000 N/m2
A.
B.
C.
D.
E.
atmospheric force on one side
1,000 N
10,000 N
10,000 N
100,000 N
100,000 N
mass exerting equivalent force
102 kg
102 kg
1020 kg
1020 kg
10200 kg
Answer: E. Force is pressure times area:
100,000 Pa x 1 m2 = 100,000 N/m2 x 1 m2 = 100,000 N
A mass of 10,200 kg (10 tons!) exerts mg = 100,000 N
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The area of your palm is 50 cm2 = 0.005 m2.
If you hold out your hand, palm up, what downward force
does the atmosphere (at about 100,000 Pa) exert on the
palm of your hand?
A.
B.
C.
D.
E.
50 N
500 N
5,000 N
50,000 N
500,000 N
Answer: B. 100,000 Pa x 0.005 m2 = 100,000 N/m2 x 0.005 m2 = 500 N
Note: a mass of 51 kg exerts about mg = 500 N = 110 pounds.
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About 10,000 kg
of air is in
a 1 m2 column
above the earth’s
surface
The weight of air:
~10,000 kg
100,000 Pa
=
=
Several miles of air = ~10,000 kg
The atmosphere exerts a force around 500 N on the palm
of your hand. Where does this force come from?
About 50 kg
of air is in
a 50 cm2 column
above the earth’s
surface
1m
1m
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Why don’t you feel 100 lbs of force on your palm?
A because you’re pushing back with 100 lbs
B because it’s pushing on all sides
C because you’re used to it
D because we’re strong
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Why don’t you feel 100 lbs of force on your palm?
Air pressure is the same all around your hand ⇒ no net force*
100 lbs downward
your hand
100 lbs upward
*Actually, there is a net force on your hand--buoyancy (the
pressure below your hand is slightly greater than the pressure
above); however, the buoyant force (in air) is negligible for objects
much denser than air. Keep watching for more on buoyancy.
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The atmosphere exerts a force around 500 N on the palm
of your hand, regardless of whether your palm is turned
upwards or downwards (or sideways).
Fluids exert the same pressure in every direction,
because fluids flow when the pressure is different in different directions.
Cube of Solid with downward pressure:
maintains shape without pressure on
sides.
Cube of Fluid with downward pressure:
flows out of shape unless an equal pressure
is applied to other sides
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Water Pressure and Gravity
Water, unlike air, maintains nearly the same density regardless of
pressure (in-compressible): 1 g/cm3, or 1000 kg/m3
Clicker Question:
A tall water trough on the surface of the
Earth, 1 m x 1m x 4m.
Each cubic meter of water weighs 10,000 N.
What is the pressure at the bottom of the
trough, 4 meters below the surface of the
water (which is at atmospheric pressure)?
A.
0
B.
40,000 Pa
C.
100,000 Pa
D.
140,000 Pa
E.
400,000 Pa
Atmospheric exerts 100,000N
pressure:
on 1 m2
100,000 Pa
10,000 N
1m
10,000 N
1m
10,000 N
1m
Answer: D. The force on the bottom square
1m
meter is 100,000 N + (4 x 10,000 N) = 140,000 N.
The pressure is 140,000 N / 1 m2 = 140,000 Pa.
10,000 N
1m
1m
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Why doesn’t water sink?
Atmospheric
exerts 100,000N
pressure:
on 1 m2
100,000 Pa
110,000 N
pressure
100,000 Pa
10,000 N
1m
No net force!
Pressure difference
exactly balances
weight!
110,000 Pa
10,000 N
1m
120,000 Pa
10,000 N
1m
130,000 Pa
10,000 N
1m
1m
140,000 Pa
1m
mg=10,000 N
120,000 N
Note: pressures from sides are
left out of drawing
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Force of water on other stuff: buoyancy
Atmospheric
pressure:
100,000 Pa
displaced
water
Remove one cubic meter of water;
replace with one cubic meter of steel.
Steel density: 8 g/cc = 8000 kg/m3
pressure
110,000 N
100,000 Pa
1m
110,000 Pa
1m
120,000 Pa
mg=80,000 N
1m
130,000 Pa
120,000 N
1m
140,000 Pa
1m
1m
1m
1m
Fnet = 70,000 N downward
--steel sinks in water
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Force of water on other stuff: buoyancy
Atmospheric
pressure:
100,000 Pa
displaced
water
Remove one cubic meter of water;
replace with one cubic meter of pine wood
Pine density: 0.5 g/cc = 500 kg/m3
pressure
What is the net force
on the cube of wood?
100,000 Pa
120,000 N
1m
110,000 Pa
1m
120,000 Pa
1m
130,000 Pa
mg= 5,000 N
1m
140,000 Pa
1m
1m
1m
1m
130,000 N
Fnet = 5,000 N upwards
--the wood floats
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Calculating buoyancy forces
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To calculate force of a fluid on an immersed object:
1. Find volume V of object (or of the part of the object
below the surface of the fluid).
2. Find the weight of a volume V of the fluid ( mg = ρVg,
where ρ is the mass density of the fluid).
3. The buoyant force of the fluid on the immersed object
is equal and opposite to the weight of the displaced
fluid.
Why? The force of the fluid on the immersed object
will be the same as the force of the fluid on the fluid
displaced by the object; that force is exactly equal to
the weight of the displaced fluid (because when the
fluid was where the object is, it was not accelerating).
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V is the volume of what?
• A the object
• B the fluid desplaced by the object
• C A and B
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Calculating buoyancy forces
•
To calculate force of a fluid on an immersed object:
1. Find volume V of object.
2. Find the weight of the same volume V of the fluid ( mg =
ρVg, where ρ is the mass density of the fluid).
3. The buoyant force of the fluid on the immersed object
is equal and opposite to the weight of the displaced
fluid.
Clicker Question :
A brick has a density of about 2000 kg/m3.
What is the net force on a brick of volume 0.001 m3 immersed in water?
(for convenience, use g~10m/s2)
A.
B.
C.
D.
E.
20 N upward
10 N upward
0
10 N downward
20 N downward
Answer: D. The weight of the brick is 20 N
(downward); the weight of 0.001 m3
displaced water is 10 N. The buoyant force
on the brick is therefore 10 N upward. The
net force on the brick is 10 N downward.
The brick sinks.
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Calculating buoyancy forces in air
•
To calculate force of a fluid on an immersed object:
1. Find volume V of object.
2. Find the weight of the same volume V of the fluid ( mg =
ρVg, where ρ is the mass density of the fluid).
3. The buoyant force of the fluid on the immersed object
is equal and opposite to the weight of the displaced
fluid.
Clicker Question :
A brick has a density of about 2000 kg/m3.
Air (at room temp/sea-level pressure) has a density around 1 kg/m3.
What is the buoyant force on a brick of volume 0.001 m3 immersed in air?
(for convenience, use g~10m/s2)
A.
B.
C.
D.
E.
0
0.0001 N upward
0.001 N upward
0.01 N upward
0.1 N upward
Answer: D. The mass of 0.001 m3 of displaced
air is 0.001 kg; its weight is 0.01 N. The
buoyant force on the brick is therefore 0.01 N
upward. The brick, which would weigh 20 N in
vacuum, appears to weigh only 19.99 N in air.
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Buoyancy depends on mass density of
object relative to mass density of fluid
• If object is less dense than fluid it floats
because the buoyancy is greater than the
gravitational force
• Examples:
Wood in water: floats
Oil in water: floats
Lead in water: sinks
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Swimmers: you can float or sink depending
on the air in your lungs
• Humans are approximately neutrally
buoyant because they are made of
water
• Jump in, at surface take in as much air
as possible - float
• Now empty your lungs all the way - sink
• Caveat: must not have clothing that
traps air bubbles
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For Stationary Water,
Pressure Changes with Height
To support the weight of the above water, the water pressure
must increase by 10,000 Pa per meter depth
(about 1/10 of an atmosphere per meter).
Water density: ρ = 1000 kg/m3
p1
Gravity
Δh
p2
1m
1m
weight = mg = ρV g = ρ (1 m2) Δh g
The pressure increase is the
increased weight
divided by the area:
p2 = p1 + mg = p1 + ρg Δh
p2 = p1 + ρg Δh
or
p2 + ρgh2 = p1 + ρgh1
or
p + ρgh = constant
(increasing h means
lowering p)
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For Stationary Water,
Pressure Changes with Height
p + ρgh = constant
Kind of like “conservation of energy” for non-moving water:
Pressure + gravitational potential energy = constant
Pressure is energy density (energy per unit volume)
Important: the difference in pressures at two points within a static
water column depends only on the differences in height of the two
points. For example, water pressure increases by about 10,000 Pa
(actually 9800 Pa) for every meter increase in depth (on the surface of
the Earth), regardless of its container.
Remember work done on body to move up a hight h.
Also important: this is true for water and most liquids; it is not true for
air and other gases. Why?
The density ρ is not constant for gases.
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For Moving Water,
Bernoulli’s Principle
p + ρgh + (1/2)ρv2 = constant
Kind of like “conservation of energy” for moving water:
Pressure + gravitational potential energy + kinetic energy = constant
Pressure is energy density (energy per unit volume)
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1
2
P?
P?
Clicker question :
a. Pressure in case 1 is larger than in case 2.
b. Pressure in case 2 is larger than in case 1.
c. Pressure in case 1 is same as in case 2.
d. Pressures can’t be determined because the
hoses are contorted into complicated shapes.
Answer: c. In both cases, the end of the hose is the same distance
below the water level. The path and shape of the hose make no
difference (in the absence of viscosity, or fluid friction). GRAVITY IS A
CONSERVATIVE FIELD; REMEMBER THE RAMP PROBLEMS
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Clicker Question
Water streams out from three spigots in an open
bucket. From which hole will the water emerge
traveling fastest?
A
B
Answer: C
The water pressure increases with depth;
a greater water pressure exerts a greater
force on the water emerging from the
spigot, causing greater acceleration,
giving the water a greater exit velocity.
C
D. A, B, and C will spout water at the same speed
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Note to Last Clicker Question
100,000 Pa
The acceleration of water as it emerges from a
spigot is governed not merely by the pressure
inside the bucket, but by the pressure difference
across the spigot.
h
100,000 Pa + ρgh
100,000 Pa
p1=100,000 Pa + ρgh
p2 = 100,000 Pa
atmospheric pressure
plus water pressure
atmospheric pressure
Δp = ρgh
resulting flow
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Speed of Water Emerging Under Pressure
Water leaves spout with kinetic energy; where does the energy come from?
A
B
C
D
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Speed of Water Emerging Under Pressure
Water leaves spout with kinetic energy; where does the energy come from?
Area
A
Δh
m
Grav. Potential Energy lost: mgh
Kinetic Energy Gained: (1/2)mv2
Energy Conservation: mgh = (1/2)mv2
h
v =
2gh
Note:
v
Volume of water leaving tank: A Δh
Mass of water leaving tank: m = ρA Δh
Δp = ρgh
Does this look familiar?
! v
Because of Δp, water exits with speed
Water exits with same same
velocity that an object would
v
gain falling (from rest) a
distance equal to the height
of the water column.
100,000 Pa
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Review
• Pressure = Force / Area
• Pressure of gases increases with density,
temperature (average kinetic energy)
• Pressure increases with depth (both for
water and for air)
• Boyancy force = weight of displaced fluid
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