Simple Harmonic Motion Chapter 15

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Lecture 22

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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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• 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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• 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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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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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 ? 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?

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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?

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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.
• .

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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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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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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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• 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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• 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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• 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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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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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

• 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