Transcript Lecture 28.BuoyancyA..

Buoyancy and
Archimedes Principle
Lecturer:
Professor Stephen T. Thornton
Reading Quiz
B) stop altogether
C) go out in a straight line
D) curve upwards
Coca-Cola
When a hole is made in the side of a
Coke can holding water, water flows
out and follows a parabolic
trajectory. If the container is dropped
in free fall, the water flow will:
A) diminish
Reading Quiz
When a hole is made in the side of a
Coke can holding water, water flows
out and follows a parabolic
trajectory. If the container is dropped
in free fall, the water flow will:
A) diminish
B) stop altogether
C) go out in a straight line
D) curve upwards
Water flows out of the hole because the
pressure outside. The water pressure is
due to the weight of the water. When the
can is in free fall, the water is weightless,
so the water pressure is zero, and hence
no water is pushed out of the hole!
Coca-Cola
water pressure inside is larger than the air
Last Time
Density, fluids
Pressure
Pressure gauges and barometers
Today
More on pressure
Pascal’s Principle
Buoyancy
Archimedes Principle
Mass flow – equation of continuity
Today we are studying fluid dynamics
after Archimedes Principle.
Measurement of Pressure;
Gauges and the Barometer
There are a number of
different types of pressure
gauges. This one is an
open-tube manometer.
The pressure in the open
end is atmospheric
pressure; the pressure
being measured will cause
the fluid to rise until the
pressures on both sides at
the same height are equal.
Tire pressure gauge.
Pascal’s Principle:
An external pressure applied to an
enclosed fluid is transmitted
unchanged to every point within
the fluid.
Do Pascal’s vases demo
Do Cartesian diver demo
A Hydraulic Lift
F1 F2
P 
A1 A2
In the hydraulic lift (show demo at
end), the liquid is enclosed, and the
pressure is the same throughout.
F1 F2
P 
A1 A2
A2
F2  F1
large increase in force
A1
But volumes are equal,
V  A1d1  A2 d 2
 A1 
d 2  d1   so d 2  d1
 A2 
Pascal’s Principle
Conceptual Quiz
Three open containers are filled with water to the A) container 1
same height and have the same surface area at
the base, but the total weight of water is different
for each. Which container has the greatest total
force acting on its base?
B) container 2
C) container 3
D) all three are equal
Conceptual Quiz
Three open containers are filled with water to the
A) container 1
same height and have the same surface area at
B) container 2
the base, but the total weight of water is different
for each. Which container has the greatest total
force acting on its base?
The pressure at the bottom of each
container depends only on the height
of water above it! This is the same for
all the containers. The total force is
the product of the pressure times the
area of the base, but since the base is
also the same for all containers, the
total force is the same.
C) container 3
D) all three are equal
Conceptual Quiz
A) water pressure
When you drink liquid through a
straw, which of the items listed is
primarily responsible for this to
function?
B) gravity
C) inertia
D) atmospheric pressure
E) mass
Conceptual Quiz
A) water pressure
When you drink liquid through a
straw, which of the items listed
below is primarily responsible for
this to function?
B) gravity
C) inertia
D) atmospheric pressure
E) mass
When you suck on a straw, you expand your lungs, which
reduces the air pressure inside your mouth to less than
atmospheric pressure. Then the atmospheric pressure pushing
on the liquid in the glass provides a net upward force on the
liquid in the straw sufficient to push the liquid up the straw.
Follow-up: Is it possible to sip liquid through a straw on the Moon?
Conceptual Quiz
You put a straw into a glass of water, place
your finger over the top so no air can get in
or out, and then lift the straw from the
liquid. You find that the straw retains some
liquid. How does the air pressure P in the
upper part compare to atmospheric
pressure PA?
A) greater than PA
B) equal to PA
C) less than PA
Conceptual Quiz
You put a straw into a glass of water, place
your finger over the top so no air can get in
or out, and then lift the straw from the
liquid. You find that the straw retains some
liquid. How does the air pressure P in the
upper part compare to atmospheric
pressure PA?
A) greater than PA
B) equal to PA
C) less than PA
Consider the forces acting at the bottom of the
straw:
PA – P – r g H = 0
This point is in equilibrium, so net force is zero.
Thus, P = PA – r g H
and so we see that
the pressure P inside the straw must be less
than the outside pressure PA.
H
Buoyant Force Due to a Fluid
P2  P1  r gL
Pressure is higher below
The buoyant force is due to the difference
in force at the bottom and the top of the
cube of sides L.
Fb  F2  F1   P2  P1  L   r gL  L  r gL
2
2
3
Fb  r gV  mg General result
Archimedes’ principle:
An object completely immersed in a fluid
experiences an upward buoyant force
equal in magnitude to the weight of fluid
displaced by the object.
Buoyant force
equals the weight of
the displaced liquid,
not the weight of the
object!
Buoyancy and Archimedes’ Principle
Archimedes’ principle: The buoyant force on an
immersed object equals the weight of displaced
fluid.
The picture below shows an object in the air,
partially submerged, and completely submerged.
Do Archimedes’ Principle I demo.
Start Archimedes’ Principle II demo.
Flotation
An object floats when it
displaces an amount of fluid
equal to its weight.
The total weight of the
block equals the weight of
the fluid displaced.
Floating an Object
That Is More Dense than
Water
The wood and water in (a)
have the same weight, and the
metal block and water in (b)
have the same weight.
Note in (c) that bowl made of
the metal floats, because the
bowl is hollow.
Volume of submerged block = Vsub
FB  mg
rfVsub g  rsVs g
Vsub
rs
 Vs
rf
weight of displaced water = weight of block
Vs  volume of solid
is water density
Remember icebergs!
Conceptual Quiz:
What happens to the water level when the ice
melts?
A) Water overflows.
B) Water level decreases.
C) Water level stays the same.
Answer: C
Water level stays the same. Because the ice cube
is floating, it displaces a volume of water equal
to its weight. When it melts it becomes water
and displaces the same volume of water it
displaced. Remember the density of ice is less
than that of water.
Conceptual Quiz:
Two identical glasses are filled to the same
level with water. One of the two glasses
has ice cubes floating in it. Which weighs
more?
A) The glass without ice cubes.
B) The glass with ice cubes.
C) The two weigh the same.
Answer: C
The ice cubes displace exactly
their own weight in water, so the
two glasses weigh the same
amount. It is essential that the
ice sticks out above the level of
the water.
Conceptual Quiz:
Now a pebble sits on top of the ice, and the
water is filled to the brim of the glass.
What happens when the ice melts?
A)
B)
C)
D)
The water overflows.
The water level decreases.
The water level stays the same.
The pebble explodes.
Answer: B
The ice makes no difference, but
now the pebble also displaces an
amount of water equal to the
pebble’s weight. When the ice
melts, the pebble drops to the
bottom and displaces a volume of
water equal to its own volume.
Because the volume of the pebble is
less than the volume of the water it
displaced originally, there is less
water displaced afterwards. The
water level drops.
Turbulent flow
Laminar flow, also
known as
streamline flow.
Consider a mass m passing through a particular
point in time t. The ratio m/ t must be
constant at every point or mass will build up.
m
 r Av makes sense, because Av is
t
volume/time.
If mass flow is constant, then
r1 A1v1  r2 A2v2
called "equation of continuity"
Gases are compressible, but liquids are not.
For liquid flow, the density r is constant.
Then A1v1 = A2 v2 for liquids.
Useful equation for many applications: for
example, water hoses.
When using a water hose, we put our thumb
over end to increase water speed.
Hose nozzle does the same thing. Decrease
area, increase speed.
Water nozzle
A1v1 = A2v2
small A, large v
Conceptual Quiz
Imagine holding two identical
bricks in place underwater. Brick
1 is just beneath the surface of
the water, and brick 2 is held
about 2 feet down. The force
needed to hold brick 2 in place is:
A) greater
B) the same
C) smaller
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2
Conceptual Quiz
Imagine holding two identical
bricks in place underwater. Brick
1 is just beneath the surface of
the water, and brick 2 is held
about 2 feet down. The force
needed to hold brick 2 in place is:
A) greater
B) the same
C) smaller
The force needed to hold the brick in
place underwater is W – FB. According
to Archimedes’ Principle, FB is equal to
the weight of the fluid displaced.
Because each brick displaces the same
amount of fluid, then FB is the same in
both cases.
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2