Physics 221, March 10
Key Concepts:
•Density and pressure
•Buoyancy
•Dynamics of ideal fluids
•What does Bernoulli's equation mean?
•Pumps and siphons
Definitions
Fluids:
Liquids  Incompressible
Gases  Compressible
Particle density:
Density:
Pressure:
ρparticle = N/V
ρ = M/V
P = F/A
Hydrostatics (liquids at rest):
Pbottom - Ptop = ρgh
Units:
1 atmosphere = 101 kPa = 14.7 pounds per square inch (psi)
Note: ρwaterg(1m) = (1000kg/m3)*(10 m/s2)*(1 m) = 10 kPa
Near the surface of the earth, the fluid at the bottom of a
container is
1. under less pressure
than the fluid on the
top.
2. under more, less, or
the same pressure as
the fluid on the top,
depending on the
circumstances.
3. under more pressure
than the fluid on the
top.
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4. under the same
1
2
3
pressure as the fluid
on the top.
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Buoyancy
An object partially or wholly immersed in a gas or liquid is acted upon by an
upward buoyant force B equal to the weight w of the gas or liquid it displaces.
B=w
Compare the densities of floating objects to the density of the liquid in which
the objects are immersed.
Will objects that float in water also float in methanol (density 790 kg/m3)?
In saturated salt water (1200kg/m3)? Explain!
Demonstration:
http://www.phys.utk.edu/demoroom/FLUIDS/POLYDENSITY.htm
Extra Credit:
The density of freshwater is
1g/cm3 and the density of
seawater is 1.03 g/cm3. A ship
will float
1. at the same level in
freshwater as in seawater.
2. higher in freshwater than in
seawater.
3. higher, lower, or at the same
level in freshwater as in
seawater, depending on its
shape.
4. lower in freshwater than in
seawater.
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You suspend a boulder weighing
fifty pounds on a rope and lower it
beneath the surface of the water in
a lake. When the boulder is fully
submerged, you find you have to
support less than 50 pounds. As
the boulder is submerged still
further, the support force needed
to hold the boulder is
1. more than just beneath the
surface.
2. the same.
3. less than just beneath the
surface.
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Extra Credit:
A block of wood of uniform density floats so that exactly half of
its volume is underwater. What is the density of the block?
1.
2.
3.
4.
0.5 kg/m3
500 kg/m3
1000 kg/m3
2000 kg/m3
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A block of wood of uniform density floats so that exactly half of its
volume is underwater. What is the density of the block?
Floating  no net force
buoyant force = weight of object
B = mobjectg = ρobject*Vobjectg
But we also have:
buoyant force B = weight w of the displaced water
B = mdisplaced_waterg
mdisplaced_water = = ρwater*½Vobject
B = ρwater*½Vobjectg
Setting the two expressions for B equal to each other:
ρobject*Vobjectg = ρwater*½Vobjectg
ρobject = ½ρwater
Ideal Fluid Dynamics
Ideal fluids  Incompressible fluids flowing without friction
equation of continuity:
Bernoulli’s equation:
(Area 1)*v1 = (Area 2)*v2
P1 + ρgh1 + ½ρv12 = P2 + ρgh2 + ½ρv22
Consequences:
Hydrostatics:
Pbottom = Ptop + ρg(htop - hbottom)
Flow in horizontal pipe:
P1 + ½ρv12 = P2 + ½ρv22
Extra Credit :
The brake system in most cars makes use of a
hydraulic system. This system consists of a fluid
filled tube connected at each end to a piston.
Assume that the piston attached to the brake pedal
has a cross sectional area of one half a square inch
and the piston attached to the brake pad has a
cross section area of two square inches. When you
apply a force of 10 pounds to the piston attached
to the brake pedal, the force at the brake pad will
be
1.
2.
3.
4.
5 pounds.
10 pounds.
20 pounds.
40 pounds.
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Water is flowing smoothly through the pipe shown in the diagram.
1.
2.
3.
4.
The pressure is lowest in section A.
The pressure is lowest in section B.
The pressure is lowest in section C.
The pressure is the same
everywhere.
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4
A gardener is watering her garden from a hose. With the water pressure full
blast holding the hose horizontally she can just reach a distance of 12 m, but
needs to water an area up to 18 m away. By what fraction must she reduce
the cross-sectional area of the hose, still keeping the hose horizontal, to be
able to water this area?
1.
2.
3.
4.
5.
½
1/3
2/3
1/6
¼
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5
A gardener is watering her garden from a hose. With the water pressure full blast
holding the hose horizontally she can just reach a distance of 12 m, but needs to water
an area up to 18 m away. By what fraction must she reduce the cross-sectional area of
the hose, still keeping the hose horizontal, to be able to water this area?
∆y = (1/2)gt2
The time it takes the water to reach the ground
depends only on the height of the outlet.
∆x = vt
The horizontal distance traveled during this time
Is proportional to v.
A1v1 = A2v2
As the area of the outlet decreases, the water speed increases.
A2 = A1v1/v2 = A1(12/18) = A1(2/3)
The cross sectional area of the hose has to be reduced by 1/3.
Water distribution
The water level in the water tower changes very slowly, v1 ~ 0.
P1 + ρgh1 + ½ρv12 = P2 + ρgh2 + ½ρv22

P1 + ρg∆h = P2 + ½ρv22 = constant
more water usage
 higher water speed
in the supply pipe
 lower water pressure
for the user
You are taking a shower in your dormitory when someone flushes a toilet nearby. The pressure in
the cold water line drops and you find yourself showering in what feels like molten lava. This loss
of cold water pressure occurs when the toilet lets cold water flow through the pipes delivering it
to the bathroom and the water’s speed in those pipes increases. Assuming all the piping to be
about on the same level, the cold water’s faster motion in the delivery pipes reduces its pressure
in the shower head because faster moving water
1.
2.
3.
4.
has less pressure than slower moving water.
has less kinetic energy than slower moving
water.
has less gravitational potential energy than
slower moving water.
cannot carry deliver much volume.
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Which one of the following is Bernoulli’s equation not involved in explaining?
1.
2.
3.
4.
5.
Why a roof can blow off a house
in a hurricane,
the buoyant force on a floating
iceberg,
dynamic lift on airplane wings,
how fast water sprays out from a
hole in a water tank,
it can explain all of these.
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Extra Credit:
The lungs can exert a negative pressure, with respect to atmospheric pressure, of up
to 1.3 kPa. To what height can you suck water through a straw?
Remember: A 1 m high column of water exerts a pressure of 10 kPa.
1.
2.
3.
4.
13 cm
1.3 m
10 m
10 mm
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