PES 2130 Fall 2014, Spendier
Lecture 29/Page 1
Lecture today: Chapter 14
1) Archimedes' Principle
2) Ideal Fluids in Motion
3) Bernoulli's Equation
Announcements:
- Exam 3 graded (current class grade written in top right corner of exam 3)
- HW11 was due Wednesday extension until Nov. Friday12pm (drop off at my
office on or before noon this Friday)
- Wednesday lecture review for FINAL (Dec 17th 5:20-7:20 )
Last time:
Fluid - Anything that can flow ==> either a liquid or a gas
pressure = force per unit area
F
N
unit  2    P  Pa.....Pascal (SI unit for pressure)
p
A
m 
true for uniform force over a flat area
Fluid pressure acts perpendicular to any surface in the fluid.
density = mass per unit volume
m
 kg 
unit  3 

V
m 
Expression for hydrostatic pressure as a function of depth or altitude.
p  p0   hg
p0 = the pressure due to 1 atmosphere
(only applicable to constant density fluids!)
Pascal's Principle
"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 its container".
PES 2130 Fall 2014, Spendier
Lecture 29/Page 2
Buoyancy DEMO Why Does Coke Sink & Diet Coke Float in Water?
Immerse sealed cans of Coke and Diet Coke in a tank of water. The can of Coke
immediately sinks, while the can of Diet Coke floats.
Buoyancy
Whether an object sinks or floats depends on its buoyancy. An object placed in water
exerts a downward force on the water. The water, however, pushes back. Archimedes’
principle states that the buoyant force exerted by water or any other fluid on an object is
equal to the weight of the water displaced by the object. If the weight of displaced water
exceeds the weight of the object, the object floats. Otherwise, it sinks. This, in part,
explains why metal ships float. It also explains why the can of Diet Coke sinks. Because
the two cans exhibit identical shapes and sizes, they displace equal amounts of water
when submerged. But the fact that the can of Coke sinks means it must weigh more than
the amount of water it displaces, whereas the can of Diet Coke weighs less.
Mathematical Explanation
Both the Diet Coke and Coke cans contain 12 fluid oz., or 355 milliliters, of liquid. Both
beverages consist primarily of water. The primary difference lies in the sweetener. Coke
contains about 325 mL of water, with a density of 1.0 g/mL, and 39 g of sugar. The
contents of the can therefore weigh 325 g + 39 g = 364 g. The can of Diet Coke,
however, contains only 0.3 g of aspartame. It therefore consists almost entirely of water
and the can’s contents therefore weigh about 355 g. This difference in weight makes the
can of Diet Coke sufficiently buoyant to float. In terms of density, the density of Diet
Coke is roughly 1.00 g/mL, the same as water. The Coke, however, exhibits a density of
1.03 g/mL.
1) Archimedes' Principle (Buoyancy)
The magnitude of the buoyant force B always equals the weight of the fluid displaced by
the object. This statement is known as Archimedes’s principle:
B = mf g
mf = mass of fluid that is displaced by body
When a body is fully or partially submerged in a fluid, a buoyant force B from the
surrounding fluid acts on the body. The force is directed upward and has a
magnitude equal to the weight mf g of the fluid that has been displaced by the body.
Fb  m f g
mf = mass of fluid that is displaced by body
PES 2130 Fall 2014, Spendier
Lecture 29/Page 3
DEMO: Cartesian Diver
Squeeze the bottle hard enough; you put pressure on the diver. That causes the air
bubble to get smaller and the entire diver to become MORE DENSE than the water
around it and the diver sinks. When you release the pressure, the bubble expands,
making the diver less dense (and more buoyant) and, alas, it floats back up.
Example:
An iceberg floating in seawater, is extremely dangerous because most of the ice is below
the surface. This hidden ice can damage a ship that is still a considerable distance from
the visible ice. What fraction of the iceberg lies below the water level?
Density of seawater, ρsw = 1030 kg/m3
Density of ice, ρi = 917 kg/m3
When an object floats, the net force on it will be zero.
Volume of submerged object displaces an amount of liquid whose weight is equal to the
weight of the object.
Aside: Ships float because they displace more water than it weights.
PES 2130 Fall 2014, Spendier
Lecture 29/Page 4
2) Ideal Fluids (Fluid flow, Continuity Equation)
Thus far, our study of fluids has been restricted to fluids at rest. We now turn our
attention to fluids in motion. The motion of fluids is a very complex phenomena. To
simplify as much as possible, we usually make three assumptions:
1. The fluid is incompressible as it flows. (Gases are hard to compress once they are
moving, so it’s a pretty good assumption for them too.) The density of an incompressible
fluid is constant.
2. The flow is steady. In steady flow, the velocity of the fluid at each point remains
constant. Steady flow is also called laminar. Non-steady is called turbulent.
3. The fluid is nonviscous. Viscosity is analogous to friction.
In a nonviscous fluid, internal friction is neglected. An object moving through the fluid
experiences no viscous force, hence fluid flow is steady.
(Demo of corn starch non-Newtonian fluid)
Corn starch demo:
Add ¼ cup of dry cornstarch to the bowl. Add about 1/8 cup (2 tablespoons, or 30 cm3)
of water to the corn starch and stir slowly. Add water slowly to the mixture, with stirring,
until all of the powder is wet.
Continue to add water until the cornstarch acts like a liquid when you stir it slowly. When
you tap on the liquid with your finger, it shouldn't splash, but rather will become hard. If
your mixture is too liquid, add more cornstarch. Your goal is to create a mixture that feels
like a stiff liquid when you stir it slowly, but feels like a solid when you tap on it with
your finger or a spoon.
Why does the cornstarch mixture behave like this?
Think of a busy sidewalk. The easiest way to get through a crowd of people is to move
slowly and find a path between people. If you just took a running start and headed
straight for the crowd of people, you would quickly slam into someone and you wouldn't
get very far. This is similar to what happens in the cornstarch mixture. The solid
cornstarch acts like a crowd of people. Pressing your finger slowly into the mixture
allows the cornstarch to move out of the way, but tapping the mixture quickly doesn't
allow the solid cornstarch particles to slide past each other and out of the way of your
finger.
We use the term “viscosity” to describe the resistance of a liquid to flow.
Water has very low viscosity!
PES 2130 Fall 2014, Spendier
Lecture 29/Page 5
Equation for the flow of an ideal fluid:
Now let's investigate the assumption that fluids are incompressible a bit further. You may
have noticed that you can increase the speed of the water emerging from a garden hose by
partially closing the hose opening with your thumb. Apparently the speed v of the water
depends on the cross-sectional area A through which the water flows.
Since we assume fluids are incompressible, as they flow an equal volume must be
moving into and out of each part of the fluid’s container. Consider the tube below with
two cylinders of fluid that have the same volume.
Setting the volumes equal
V1  V2
A1 x1  A2 x2
and dividing by time
x
x
A1 1  A2 2
t1
t2
gives relation between speed and cross-sectional area which is called the Continuity
Equation for the flow of an ideal fluid:
A1v1  A2 v2
It tells us that the flow speed increases when we decrease the cross-sectional area
through which the fluid flows.
PES 2130 Fall 2014, Spendier
Lecture 29/Page 6
3) Bernoulli's Equation
Now let’s add elevation change to the tube, i.e. a tube that connects a lower point and a
higher point in space. As a fluid moves through a region where its speed and/or elevation
above the Earth’s surface changes, the pressure in the fluid varies with these changes.
The relationship between fluid speed, pressure, and elevation was first derived in 1738 by
the Swiss physicist Daniel Bernoulli.
Consider the case of water flowing through a smooth pipe.
The Bernoulli Equation is derived from conservation of energy and work-energy ideas
that come from Newton's Laws of Motion. (Look in book for derivation/proof)
Let y1, v1, and p1 be the elevation, speed, and pressure of the fluid entering at the left, and
y2, v2, and p2 be the corresponding quantities for the fluid emerging at the right. By
applying the principle of conservation of energy to the fluid, can show that these
quantities are related by
or
Bernoulli’s equation is strictly valid only to the extent that the fluid is ideal. If viscous
forces are present, thermal energy will be involved.
Check: Let us apply Bernoulli’s equation to fluids at rest, by putting v1 = v2 = 0
Which we derived already in Lecture 27!
PES 2130 Fall 2014, Spendier
Lecture 29/Page 7
Example:
Water enters a house through a pipe with an inside diameter of 2.0 cm at an absolute
pressure of 4.0 x 105 Pa (about 4 atm). A 1.0 cm 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
i) the flow speed
ii) and pressure,
in the bathroom.
DEMO: cornstarch in water grows weird shapes due to vibrations of a speaker
Non-Newtonian Fluid (cornstarch and water). The cornstarch acts almost like a solid
when its impacted quickly  fast vibrations of a speaker cone makes the fluid squirm
around.
https://www.youtube.com/watch?v=SYMvOxIsES4