Vibrations and Waves

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Vibrations and Waves

• Chapter 11
• (2008-2009)

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• A repeating back-and-forth motion about an
  equilibrium position is a vibration.
• A disturbance that is transmitted progressively
  from one place to the next with no actual
  transport of matter is a wave.
• Light and sound are both forms of energy that
  move through space as waves.

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Vibrations and Waves

• Vibrating spring Change in velocity and
  restoring force
• Pendulum Change in velocity and restoring force

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Simple Harmonic Motion (SHM)

• Hookes Law periodic motion, back and forth over
  the same path
• Mass attached to spring or wrecking ball swinging
  back and forth
• At equilibrium position (?x0) velocity is
  maximum
• Restoring force and acceleration0
• ?x gt0 spring exerts a force on the mass toward
  the equilibrium position

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• As ?x decreases so does the restoring force and
  velocity increases
• The masss momentum causes it to over shoot the
  equilibrium position
• Therefore the spring stretches past equilibrium
  and spring force and acceleration reach maximum
• Restoring force and acceleration is proportional
  yet opposite to ?x

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• When ?x, F, and acceleration is maximum, velocity
  0
• Damping causes the spring to come to rest
• SHM vibration about an equilibrium position in
  which a restoring force is proportional to the
  displacement from equilibrium

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• Hookes Law Felastic -kx
• (-) signifies the direction of the spring force
  is always opposite direction of masss
  displacement
• F spring force (N), k spring constant (N/m), x
  is displacement (m)
• Fnet Felastic Fg
• Fg -mg ? negative due to downward vector
  position and opposite spring force

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Board Work

• A 76 N crate is attached to a spring (k 450
  N/m). How much displacement is caused by the
  weight of this crate?
• A spring (k1962 N/m) loses its elasticity if
  stretched more than 50.0 cm. What mass of the
  heaviest object can the spring support with out
  being damaged?

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Elastic Potential Energy

• A stretched or compressed spring has elastic
  potential energy
• Bow and arrow
• PEelastic KE of arrow when released
• B/c ME must be conserved, KE of bow, arrow, and
  bowstring is equal to PEelastic stored in the bow
• Spring PE is elastic
• Pendulum PE is gravitational
• MEtotal PE KE and is constant
• Neglect friction

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The Simple Pendulum

• Bob mass centered at one pt.
• Mass of string is negligible
• The restoring force of a pendulum is a component
  of the bobs weight
• Two forces acting on pendulum are Ft and Fg
• Fg can be resolved into its two components of Fx
  and Fy

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• Fgx (perpendicular to string) only force acting
  on the bob in direction of motion
• Fgx always pushes or pulls the bob toward
  equilibrium restoring force
• For small angles the pendulums motion is SHM
  (lt15o)
• The restoring force is proportional to the
  displacement

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Amplitude, Period, Frequency

• Amplitude maximum displacement from an
  equilibrium pt related to energy
• Pendulum angle between max. displacement and
  equil position
• Spring max amount of stretch or compression from
  equil position

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Complete Cycle

• Max displacement on both sides of a swinging or
  vibrating system
• Period and frequency measures time
• Period (T) time it takes to complete one cycle
  of motion (s/cycle)
• Period is back and forth motion
• Frequency (f) of cycles per unit of time,
  usually one second
• Cycles/s, 1/s, Hertz
• Period and frequency are inversely related
• f 1/T ? T 1/f

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Vibration of a Pendulum

• Two pendulums of the same length have the same
  period regardless of mass.

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• Period depends on string length and free-fall
  acceleration
• For small amplitudes the period of pendulum does
  not depend on amplitude
• Changing length of pendulum does affect its
  period
• Amplitude must be less than about 15o in order
  for its motion to be simple harmonic motion

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Board Work

• What is the period of a 3.98 m long pendulum? A
  99.4 cm long pendulum?
• A desk top toy swings back and forth once every
  1.0 s. How tall is the toy?
• What is the free fall acceleration at a location
  where a 6.00 m long pendulum swings through
  exactly 100 cycles in 492 s?

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Mass-Spring System

• A weight attached to a spring undergoes simple
  harmonic motion.

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• Period depends on the mass and spring constant
• Hookes Law Felastic -kx
• ?x proportional to Felastic
• Mass does not effect the restoring force
• Heavier masses attached to spring increases the
  inertia w/out providing a compensating increase
  in restoring force

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• Increase inertia, increase mass ? smaller
  acceleration
• Increase mass increase T
• Increase k (stiffer spring) increase force to
  stretch spring
• Increase k, decrease T
• Period of a mass spring system in SHM

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Board Work

• A 1.0 kg mass attached to one end of a spring
  completes one oscillation every 2.0 s. K?
• What size mass will make the spring vibrate once
  every 1.0 s?

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Wave Motion

• Water waves move from one place to another, but
  the water itself is not carried with it
• Wave the motion of a disturbance
• Mechanical Wave movement of energy without a net
  movement of matter

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Wave Motion

• Needs a medium to transfer energy
• Particles vibrate around an equilibrium position
• Medium material through which a disturbance
  travels

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Electromagnetic Waves

• Do not need a medium to travel through
• Light will travel through a vacuum

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Wave Types

• Pulse Wave single nonperiodic disturbance
• Periodic Wave wave whose source is some form of
  periodic motion
• Transverse Wave wave motion perpendicular to
  direction of energy
• Longitudinal Wave motion parallel to direction
  of energy
• Density or compression waves
• Compression (crest) rarefraction (trough)

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Transverse Wave

• Crest highest point above the equilibrium
  position
• Trough lowest pt
• Wavelength (?) distance between 2 adj. Similar
  pts of the wave (m)

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Period, Frequency, and Speed

• Wave motion generated by a vibrating object
• Frequency (f) describes the of crests or
  troughs that pass a given pt in unit of time
  (usually seconds) (1/s)
• Period (T) time for one cycle to pass a given
  point
• Period inversely proportional to frequency
• V ?/T ?f and is constant for a given medium

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Board Work

• A 2640 Hz whistle produces a sound wave that has
  a wavelength of 50.0 cm in water. What is the
  speed of sound in water?

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Energy

• Waves transfer energy without transferring the
  medium
• Energy depends on the amplitude
• Greater then amplitude ? greater the energy
• E A2

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Wave Interaction

• Two different material objects can never occupy
  the same space at once
• Mechanical waves are not matter, but displacement
  of matter
• Two waves can occupy the same space at once
  superposition
• Form interference pattern
• 2 waves are the sum of their amplitudes
• Constructive Fnet increases
• Destructive Fnet decreases

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Superposition

• Constructive or destructive interference

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Reflection at Free End

• Free boundary reflected wave not inverted
• Reflection reflected wave is identical to the
  incident pulse

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Reflection

• Fixed boundary reflected wave inverted

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Transmitted

• Wave at new medium, part is transmitted and part
  is reflected
• The reflected portion of the wave is always
  inverted
• Less dense medium transmitted wave not inverted
  and moves faster
• More dense medium transmitted wave not inverted
  but moves slower

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Transmitted
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Standing Waves

• Wave pattern that results when two waves of same
  frequency, ?, and amplitude travel in opposite
  directions and interfere
• Wave appears to stand still
• Standing waves have nodes and antinodes
• (N) node point at which two waves cancel
• (A) antinodes midway through nodes with largest
  amplitude

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Standing Waves