Vibrations

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Vibrations

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Title: Vibrations


1
Vibrations Waves
  • Chapter 11

2
11.1 Simple Harmonic Motion
  • Consider a mass upon a frictionless surface and
    attached to a spring
  • Define the unstretched position of the mass as
    equilibrium position
  • When the spring is stretched or compressed and
    then released, the mass vibrates back forth
  • Block-spring system simulation

3
  • Velocity is max at equilibrium (displacement 0)
  • Max displacement, force, and acceleration occur
    at maximum displacement

4
Simple Harmonic Motion
  • As mass moves from equilibrium a restoring force
    returns it toward equilibrium
  • SHM is vibration about an equilibrium position in
    which the restoring force is proportional to the
    displacement from equilibrium

5
Hookes Law
  • Felastic -kx
  • k is the spring constant
  • x is displacement
  • Recall similarity of elastic potential energy
    PEelastic ½ kx2
  • A stretched or compressed spring has potential
    energy

6
Sample problem A
  • A mass of 0.55 kg is attached to a vertical
    spring, which displaces the spring 2.0 cm from
    its equilibrium position. Find the spring
    constant.
  • Since there is no vibration, a static equilibrium
    exists, i.e. Fg Felastic
  • Fg mg 0.55 kg x 9.81m/s2 -5.4N
  • Fe -kx
  • -k Fe/x
  • k -5.4N/-0.02m
  • k 270 N/m

7
Simple Pendulum
  • Definitions assumptions
  • Bob is a point mass at the end of a string
  • Mass of string air resistance are negligible
    (have no effect on operation of pendulum)
  • For small angles of excursion (15), pendulum
    operates as SHM
  • Pendulum simulation

8
Pendulum SHM
  • Restoring force is a component of the bobs
    weight (Fig 12.6)
  • x-component of bobs weight pulls bob back toward
    equilibrium position
  • When discounting friction, total mechanical
    energy is constant during operation of pendulum
  • ME KE PE ½ mv2 mgh
  • Fig 12.7 Table 12-1

9
Conservation of Mechanical Energy in SHM
10
Displacement, velocity, acceleration, and force
in SHM
11
Graphical Relationship of Position, Velocity, and
Acceleration in SHM
y A cos(2pft f)
Note the relationship of maximum, minimum, and
zero values of these quantities
http//www.unistudyguides.com/wiki/Oscillations
12
11.2 Measuring SHM
  • Measures of SHM include
  • Amplitude
  • Maximum displacement from equilibrium
  • Period (T)
  • Time required for one cycle of SHM
  • Frequency (f)
  • Number of vibrations or cycles per unit time

13
Relationship of Period and Frequency
  • Period and frequency are inversely related

14
Period of a Simple Pendulum
  • Depends upon the length of string and
    acceleration of gravity
  • Where T is the period and L is the length of the
    string

15
Sample B
  • A pendulum extends from the ceiling. If the bob
    nearly touches the floor, and its period is 12 s,
    how high is the ceiling?
  • T 12 s g 9.81 m/s2
  • Unknown L

16
Period of a mass-spring system
  • Period of a mass-spring system simulation

17
Period of a mass-spring system
  • Depends on mass and the spring constant
  • m is mass (kg)
  • k is the spring constant (N/m)

18
11.3 Properties of WavesObjectives
  • Distinguish local particle vibrations from
    overall wave motion
  • Differentiate between pulse waves and periodic
    waves
  • Interpret wave forms of transverse and
    longitudinal waves
  • Apply the relationship among wave speed,
    frequency, and wavelength to solve problems
  • Relate energy to amplitude

19
Wave motion
  • Waves involve the vibration of matter
  • Waves, then, are the motion (propagation) of a
    disturbance in matter
  • Medium the material through which the
    disturbance travels, e.g. water
  • Mechanical Wave a wave that requires a physical
    medium for propagation

20
Wave Types
  • Pulse wave
  • A single non-periodic disturbance
  • Periodic wave
  • A wave whose source is some periodic motion
  • Phet Wave Simulator

21
Wave Types
  • Waves can also be classified according to the
    direction of the vibration relative to the
    direction of the wave
  • Transverse wave
  • Particles vibrate perpendicularly to the
    direction of the wave
  • Longitudinal (compression) wave
  • Particles vibrate parallel to the direction of
    the wave
  • Longitudinal and Transverse Wave Motion

22
Crest, Trough, Wavelength in a Longitudinal wave
  • Correspond to density of the medium during wave
    cycles

23
Crest, Trough, Wavelength
  • Both transverse and longitudinal waves can be
    described with a crest, trough, and wavelength
  • Fig 12.13
  • Fig 12.15

24
Period, Frequency, Wave Speed
  • Period the time for one complete cycle of
    vibration
  • Period time for one wavelength to pass a given
    point
  • Frequency number of wave cycles per second
  • Units of s-1 or Hz

25
Speed of a wave
26
Waves Transfer Energy
  • A disturbance is caused by energy
  • E.g. the energy from a collision of a stone with
    water is transmitted through waves in the water
  • Energy of a wave is related to its amplitude
  • E a2
  • Wave damping loss of energy through time
  • Wave Simulator

27
11.4 Wave InteractionsObjectives
  • Apply the superposition principle
  • Distinguish constructive and destructive
    interference
  • Predict when reflected waves will be inverted
  • Distinguish between traveling waves and standing
    waves
  • Identify nodes and antinodes of a standing wave

28
Wave Interference
  • Waves on a String (http//phet.colorado.edu/sims/w
    ave-on-a-string/wave-on-a-string_en.html
  • Superposition two waves can occupy the same
    space at the same time
  • When two waves occupy the same space at the same
    time, the resultant wave is the sum of the two
    waves
  • Constructive interference displacements of the
    two waves are on the same side of equilibrium,
    producing greater amplitude
  • Destructive interference displacements of the
    two waves are on opposite sides of equilibrium,
    reducing amplitude

29
Reflection
  • Waves on a String (http//phet.colorado.edu/sims/w
    ave-on-a-string/wave-on-a-string_en.html
  • Waves are reflected when they encounter a
    boundary
  • At a free boundary, incident (incoming) and
    reflected pulses occur on the same side of
    equilibrium
  • At a fixed boundary, incident and reflected
    pulses occur on opposite sides of equilibrium

30
Standing Waves
  • Standing wave a wave pattern that results when
    two waves of the same frequency, wavelength, and
    amplitude travel in opposite directions and
    interfere
  • Node represents complete destructive
    interference and is therefore no displacement (is
    stationary)
  • Antinode mid-way between two nodes, maximum
    displacement occurs
  • Fig 12-22, p. 463

31
Nodes and Antinodes
Node Complete destructive interference
stationary zero displacement Antinode Midway
between nodes, maximum amplitude maximum
displacement
Standing wave - Wikipedia, the free encyclopedia
32
A Standing Wave is a Stationary Wave
  • A standing wave is a wave that remains in a
    constant position.
  • The wave is constrained by two fixed boundaries,
    forming nodes
  • Results from interference of two waves traveling
    in opposite directions.
  • There is no net propagation of energy.

33
Wavelengths of standing waves
  • Because each end of the wave must be a node, only
    certain frequencies produce standing waves
  • Consider a string of length L
  • Describe wavelength in terms of L
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