Announcements

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Lecture 22

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Announcements

• Homework due TODAY at the end of lecture
• Homework 12 based on ch 12 13, due on Dec 5th.
• I may post some practice problem suggestions for
  ch 14 and 15.
• Final Exam on Thursday, Dec 13, 9am-12pm, in
  BARHOL 168
• Close book, close lecture notes
• Chapters 1-15, with emphasis on post midterm 2
  material
• Emphasis of the exam problem solving (6-8
  problems)
• Simple calculator and one-page formula sheet
  allowed
• Questions? Suggestions?

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Simple Harmonic MotionChapter 15
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• A block of mass 1 kg is attached to a spring with
  k100 N/m and free to move on a frictionless
  horizontal surface. At t0, the spring is
  extended 5 cm beyond its equilibrium position and
  the block is moving to the left with a speed of
  1 m/s. What is the displacement of the block as a
  function of time?

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What are the values of A, ?, and ? ?

• A, ? are determined by initial conditions (x,v at
  t0)
• they are not properties of the oscillating
  system.

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• Three 10,000 kg ore cars are held at rest on a
  30o incline using a cable. The cable stretches 15
  cm just before a coupling breaks detaching one of
  the cars. Find (a) the frequency of the resulting
  oscillation of the remaining two cars and (b) the
  amplitude of the oscillation.

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m 10000 kg d 0.15 m first compute spring
constant of rope
then compute frequency of oscillation
the amplitude equals the original displacement
from the equilibrium position of two cars on the
rope A d ? d0 0.05 m
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• A 10 g bullet strikes a 1 kg pendulum bob that is
  suspended from a 10 m long string. After the
  collision the two objects stick together and the
  pendulum swings with an amplitude of 10o.
• What was the speed of the bullet when it hit the
  bob?

10o
10 g
v?
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Energy of simple pendulum
using
with
Get velocity of bob at equilibrium point from
total energy
get velocity of bullet from momentum conservation
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CO2 molecule

• Carbon dioxide is a linear molecule. The C-O
  bonds act like springs. This molecule can vibrate
  such that the oxygen atoms move symmetrically in
  and out, while the carbon atom is at rest. The
  frequency of this vibration is observed to be
  2.83x1013 Hz.
• What is the spring constant of the C-O bond?
• In which other way can the molecule vibrate and
  at what frequency?
• If the amplitude is the same, in which mode is
  the energy of the molecule higher?

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CO2 molecule

• a)
• b)
• c)
• ? the energy is the same for both modes.

Parallel springs See Lecture 10
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• Which of the following spring arrangements will
  oscillate with the smallest angular frequency?
  Assume that all springs are identical.
• (1) (2) (3) (4)

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Driven Harmonic Oscillator

• What happens if you have an oscillator, such as a
  mass on a spring, where an external force is
  acting on the system?
• Example Motion of a building or bridge during an
  earthquake
• Essentially all objects have one or more natural
  frequencies that they will oscillate at if they
  are initially displaced from equilibrium
• Example mass on a spring has a natural frequency
  given by
• If an oscillating external force is applied with
  angular frequency w close to the natural
  frequency w0, the results can be dramatic

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Driven Harmonic Oscillator

• To get motion, use Newtons 2nd law
• Suppose the external force is sinusoidal
• Eventually, the objects motion will oscillate
  with frequency w since thats the frequency of
  the applied force

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Driven Harmonic Oscillator

• Solution for driven harmonic oscillator is
  somewhat more complicated than what we have done
  so far in Physics 5
• With some effort, one can solve the equation of
  motion
• where the frequency and amplitude are given by

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Driven Harmonic Oscillator

• Example Mass on spring
• Notice how amplitude and phase change as the
  frequency of the external force crosses the
  natural frequency
• Example Vibrations of a solid object
• Solid objects typically have one or more natural
  frequencies that they oscillate at
• If the damping is small, large oscillations occur
  when driven at these natural frequencies
• These vibrations can be a considerable source of
  stress on the object!

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Oscillations Summary

• We get periodic motion when force acts to push
  object back towards equilibrium position
• Many problems exhibit simple harmonic motion
• Energy exchanged between kinetic and potential
  energy, total mech. energy unchanged for undamped
  oscillations
• Correspondence between simple harmonic motion and
  uniform circular motion
• Amplitude of oscillations decays with a damping
  force
• Driven oscillations exhibit a resonance at the
  natural frequency
• (ex Tacoma Narrows bridge collapse video)

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Fluids Chapter 14
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Fluids

• What is a fluid?
• A substance that flows
• Examples include liquid, gas, plasma, etc
• A simple fluid can withstand pressure but not
  shear
• Density
• Density
• Unit kg/m3
• Examples density of water 1000 kg/m3 air 1.21
  kg/m3
• A materials specific gravity is the ratio of the
  density of the material to the density of water
  at 4C.
• What is special about water at 4C?
• Water is most dense at that temperature.
• Aluminum has a specific gravity of 2.7 it is
  2.7 times more denser than water at 4C.

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A table of densities
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Pressure

• Pressure
• Unit pascal (Pa)N/m2
• Examples 1 atm1.01x105 Pa760 torr14.7 lb/in2
• Torricelli (torr) is defined as the pressure of
  mm Hg. Blood pressure 70/120 torr
• At any point in a fluid at rest, the pressure is
  the same in all directions.
• If this were not true there would be a net force
  on the fluid and it could not be at rest.
• The force due to fluid pressure acts
  perpendicular to any surface.
• Else there would be a force component along the
  surface which would accelerate the fluid.

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Atmospheric pressure

• At atmospheric pressure, every square meter has a
  force of 100,000 N exerted on it, coming from air
  molecules bouncing off it!
• Why dont we, and other things, collapse because
  of this pressure?
• We have an internal pressure of 1 atmosphere.
• Objects like tables do not collapse because
  forces on top surfaces are balanced by forces on
  bottom surfaces, etc.

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Fluid-Statics

• Static equilibrium
• A simpler expression
• Where p0 is the pressure at the surface, and h is
  depth of the liquid
• The pressure at any point in a fluid is
  determined by the density of the fluid and the
  depth. It does not depend on any horizontal
  dimension of the fluid or its container. It also
  does not depend on the shape of the container.
  (example pascal vases)

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Measuring pressure

• The relationship between pressure and depth is
  exploited in manometers (or barometers) that
  measure pressure.
• A standard barometer is a tube with one end
  sealed.
• The sealed end is close to zero pressure, while
  the other end is open to the atmosphere.
• The pressure difference between the two ends of
  the tube can maintain a column of fluid in the
  tube, with the height of the column being
  proportional to the pressure difference.
• pressure at bottom of column atmospheric
  pressure

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Mercury/Water barometer

• Mercury
• atmospheric pressure pushes Hg column up ? unit
  mm-Hg (torr)
• Thus Atmospheric pressure pushes the Hg column
  up by
• 101.3 kPa/133 Pa/mm 760 mm
• Water
• Thus atmospheric pressure pushes the water column
  up by
• 101.3 kPa/9.8 Pa/mm 10.3 m
• another unit 1 bar 105 N/m2
• in calculations only use N/m2 Pa (SI unit)

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Gauge Pressure

• Gauges measure pressure relative to atmospheric
  pressure absolute pressure gauge pressure
  atmospheric pressure
• manometer (height of column of liquid measures
  gauge pressure)

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Blood pressure

• A typical reading for blood pressure is 120 over
  80.
• What do the two numbers represent?
• What units are they in?
• 120 mm Hg (millimeters of mercury) is a typical
  systolic pressure, the pressure when the heart
  contracts.
• 80 mm Hg is a typical diastolic pressure, the
  blood pressure when the heart relaxes after a
  contraction.
• 760 mm Hg is typical atmospheric pressure. The
  blood pressure readings represent gauge pressure,
  not absolute pressure they tell us how much
  above atmospheric pressure the blood pressure is.

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• A delicious drink sits on the
  patio. From your balcony several stories up you
  manage to lower a
  straw into the glass, which is 15
  m below you. Can you syphon up the drink?

15 m

• yes, but I will have to suck really hard
• probably not, but a vacuum pump could
• no, this is not possible
• I dont know

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Water pressure

• At the surface of a body of water, the pressure
  you experience is atmospheric pressure. Estimate
  how deep you have to dive to experience a
  pressure of 2 atmospheres.
• h works out to 10 m. Every 10 m down in water
  increases the pressure by 1 atmosphere.

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Rank by pressure

• A container, closed on the right side but open to
  the atmosphere on the left, is almost completely
  filled with water, as shown. Three points are
  marked in the container. Rank these according to
  the pressure at the points, from highest pressure
  to lowest.
• A B gt C
• B gt A gt C
• B gt A C
• C gt B gt A
• C gt A B
• some other order

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Blaise Pascal (1623-1662)

• A change in the pressure applied to an enclosed
  incompressible fluid is transmitted undiminished
  everywhere in the fluid and to the walls of the
  container

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• A container is filled with oil and fitted on both
  ends with pistons. The area of the left piston is
  10 mm2 that of the right piston is 10,000 mm2.
  What force must be exerted on the left piston to
  keep the 10,000 N car on the right at the same
  height?

10000 N

• 10 N
• 100 N
• 10,000 N
• 106 N
• 108 N

?
10 mm2
10000 mm2
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Pascals Principle

• Hydraulic lever (see diagram on right)
• Something has to give
• Since the liquid is incompressible, the volume
  drop on the left is equal to the volume increase
  on the right, ie.

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• Pascal placed a long thin tube vertically into a
  wine barrel. When the barrel and tube were filled
  with water to a height of 12 m, the barrel burst.
  
• (a) what is the mass of the water in the tube?
• (b) what is the net force exerted onto the lid of
  the barrel?

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The Buoyant Force

• With fluids, we bring in a new force.
• The buoyant force is generally an upward force
  exerted by a fluid on an object that is either
  fully or partly immersed in that fluid.
• Lets survey your initial ideas about the buoyant
  force.

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The Buoyant Force

• A wooden block with a weight of 100 N floats
  exactly 50 submerged in a particular fluid. The
  upward buoyant force exerted on the block by the
  fluid
• has a magnitude of 100 N
• has a magnitude of 50 N
• depends on the density of the fluid
• depends on the density of the block
• depends on both the density of the fluid and
  the density of the block

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Learning by Analogy

• Our 100 N block is at rest on a flat table. What
  is the normal force exerted on the block by the
  table?
• To answer this, we apply Newtons Second Law.
  There is no acceleration, so the forces balance.

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Apply this to Buoyant force

• Apply the same method when the block floats in
  the fluid.
• What is the magnitude of the buoyant force acting
  on the block?
• To answer this, we apply Newtons Second Law.
  There is no acceleration, so the forces balance.

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Reviewing the normal force

• We stack a 50-newton weight on top of the 100 N
  block. What is the normal force exerted on the
  block by the table?
• To answer this, we apply Newtons Second Law.
  There is no acceleration, so the forces balance.
  The block presses down farther into the table
  (this is hard to see).

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Buoyant force

• We stack a 50-newton weight on top of the 100 N
  block. What is the buoyant force exerted on the
  block by the fluid?
• To answer this, we apply Newtons Second Law.
  There is no acceleration, so the forces balance.
  The block presses down farther into the fluid
  (this is easy to see).

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Apply Newtons Second Law

• Even though we are dealing with a new topic,
  fluids, we can still apply Newtons second law to
  find the buoyant force.

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Three Beakers

• The wooden block, with a weight of 100 N, floats
  in all three of the following cases, but a
  different percentage of the block is submerged in
  each case. In which case does the block
  experience the largest buoyant force?
• 4. The buoyant force is equal in all three
  cases.

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Three Beakers

• What does the free-body diagram of the block look
  like?
• What is the difference between these fluids?
• The density

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• A block of weight mg 45.0 N has part of its
  volume submerged in a beaker of water. The block
  is partially supported by a string of fixed
  length. When 80.0 of the blocks volume is
  submerged, the tension in the string is 5.00 N.
  What is the magnitude of the buoyant force acting
  on the block?

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Apply Newtons Second Law

• The block is in equilibrium all the forces
  balance.
• Taking up to be positive

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• Water is steadily removed from the beaker,
  causing the block to become less submerged. The
  string breaks when its tension exceeds 35.0 N.
  What percent of the blocks volume is submerged
  at the moment the string breaks?

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Apply Newtons Second Law

• The block is in equilibrium
• all the forces balance.
• Taking up to be positive
• The buoyant force is proportional to the volume
  of fluid displaced by the block. If the buoyant
  force is 40 N when 80 of the block is submerged,
  when the buoyant force is 10 N we must have 20
  of the block submerged.

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• After the string breaks and the block comes to a
  new equilibrium position in the beaker, what
  percent of the blocks volume is submerged?
• what does the free-body diagram look like now?

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Apply Newtons Second Law

• The block is in equilibrium
• all the forces balance.
• Taking up to be positive
• The buoyant force is proportional to the volume
  of fluid displaced by the block. If the buoyant
  force is 40 N when 80 of the block is submerged,
  when the buoyant force is 45 N we must have 90
  of the block submerged.

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Archimedes Principle

• While it is true that the buoyant force acting on
  an object is proportional to the volume of fluid
  displaced by that object.
• Example cartesian diver
• But, we can say more than that. The buoyant force
  acting on an object is equal to the weight of
  fluid displaced by that object. This is
  Archimedes Principle.

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A Floating Object

• When an object floats in a fluid, the downward
  force of gravity acting on the object is balanced
  by the upward buoyant force.
• Looking at the fraction of the object submerged
  in the fluid tells us how the density of the
  object compares to that of the fluid. (example
  density blocks, coke cans)

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Beaker on a Balance

• A beaker of water sits on a scale. If you dip
  your little finger into the water, what happens
  to the scale reading? Assume that no water spills
  from the beaker in this process.
• 1. The scale reading goes up
• 2. The scale reading goes down
• 3. The scale reading stays the same

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Three Blocks

• We have three cubes of identical volume but
  different density. We also have a container of
  fluid. The density of Cube A is less than the
  density of the fluid the density of Cube B is
  exactly equal to the density of the fluid and
  the density of Cube C is greater than the density
  of the fluid. When these objects are all
  completely submerged in the fluid, as shown,
• Which object has the largest buoyant force acting
  on it?
• 1. Cube A
• 2. Cube B
• 3. Cube C
• 4. The cubes have equal buoyant forces

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Three Blocks

• Each cube displaces an equal volume of the same
  fluid, so the buoyant force is the same on each.

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• Two identical glasses are filled to the brim with
  water. One of the two glasses has a ball floating
  in it. Which glass weighs more?

• The glass without the ball
• The glass with the ball
• The two weigh the same

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• A boat carrying a large boulder is floating in a
  lake. The boulder is thrown overboard and sinks.
  What happens to the water level in the lake
  (relative to the shore)?

• it sinks
• it rises
• it remains the same

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• Cartesian diver
• The diver is an object in a sealed container of
  water.
• Air in the diver makes it buoyant enough to
  barely float at the water's surface.
• When the container is squeezed, the pressure
  compresses the air and reduces its volume. This
  permits more water to enter the diver, resulting
  in it being less buoyant and sinking.
• regular coke and diet coke

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The origin of the buoyant force

• The net upward buoyant force is the vector sum of
  the various forces from the fluid pressure.
• Because the fluid pressure increases with depth,
  the upward force on the bottom surface is larger
  than the downward force on the upper surface of
  the immersed object.

• This is for a fully immersed object. For a
  floating object, h is the height below the water
  level, so we get

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When the object goes deeper

• An object is totally immersed in a fluid. If we
  displace the object immersed deeper into the
  fluid, what happens to the buoyant force acting
  on it? Assume the fluid density is the same at
  all depths. The buoyant force
• increases
• decreases
• stays the same

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When the object goes deeper

• If the fluid density does not change with depth,
  all the forces increase by the same amount,
  leaving the buoyant force unchanged!

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Archimedes Principle

• Buoyant force
• Objects that float
• Dry wood, ice, some plastics, oil, wax
  (candles)
• Boats made of woods, ceramic, steel, or any other
  materials, as long as they are hollow enough
• Objects that sink
• Rocks, sands, clay, metal, etc.
• Any material with density larger than water
• Apparent weight (example submerged object weighs
  less)
• Apparent weightactual weight - buoyant force
• What is your apparent weight in water? (no more
  than a few pounds!)

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Unbalancing the forces

• If we remove the balance between forces, we can
  produce some interesting effects. Demonstrations
  of this include
• The Magdeburg hemispheres (see below)
• Crushing a can

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Crush a can

• Remember that this is just the collective effect
  of a bunch of air molecules!

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Summary

• Density and pressure of fluids
• Air pressure, blood pressure and underwater
  pressure
• Pascals Principle
• Archimedes Principle