Vibrations

1
Vibrations Waves

• Chapter 11

2
Simple Harmonic Motion

• Periodic motion repeated motion
• Good example of periodic motion is mass on a
  spring on a frictionless surface

3
Points of mass on a spring

• Force is always toward equilibrium
• Force acceleration are zero at equilibrium
• Speed is greatest at equilibrium
• Momentum of mass continues it moving past
  equilibrium
• Force acceleration are greatest at maximum
  displacement
• Force is referred to as a restoring force
• Damping friction slows movement of mass-spring
  system

4
Hookes Law

• Restoring force is directly proportional to
  displacement of mass
• Felastic -kx
• Negative shows that force is always opposite
  direction of displacement
• k is the spring constant units are N/m
• Greater the k the stiffer the spring

5
Simple Harmonic Motion

• Describes any periodic motion that is the result
  of a restoring force that is proportional to
  displacement
• Always a back and forth motion over the same path

6
Springs Elastic PE

• When you stretch a spring, you store energy
• This energy is elastic PE
• When spring is released, object attached to it
  has KE
• This energy is conserved

7
Simple Pendulums

• Applies only for angles under 15o
• Mass called a bob attached to end of massless
  string
• X-component of gravity is restoring force of a
  pendulum y-component tension of string balance
  out
• PEg increases as pendulums displacement increases

8
Internet Sites

• Mass on a Spring
• Mass on a Spring 2

9
Measuring Simple Harmonic Motion

• Amplitude max displacement from equilibrium
  position angle in pendulums amount of
  compression/stretch in spring
• Period (T) time for one complete cycle
• Frequency (f) of cycles in a unit of time
  cycles per second measured in Hertz (Hz)
• T and f are reciprocals of each other

10
Factors affecting Period of a Pendulum

• Period of pendulum depends on length of string
  and gravity
• For small amplitudes, mass amplitude do not
  affect period
• Formula on bottom of p. 377

11
Period of mass-spring system

• Period of mass-spring depends on mass of object
  spring constant
• Restoring force in Hookes law is dependent on
  displacement, not mass
• Increasing mass on spring increases inertia but
  not restoring force
• Heavier masses have longer periods
• Larger spring constants provide greater force,
  greater acceleration, shorter periods
• Formula on page 380

12
Properties of Waves

• Medium is not carried along with a wave
• Particles of medium are disturbed but return to
  original position after energy passes through
• Medium material wave disturbance travels
  through
• Mechanical waves require a medium

13
Wave Types

• Pulse waves are composed of a single pulse
• If source of a wave is undergoing SHM, it
  produces a sine wave
• Transverse waves have medium moving at right
  angles to the direction the wave is moving

14
Parts of a Wave

• Crest highest pt above equilibrium
• Trough lowest pt below equilibrium
• Amplitude distance from equilibrium to crest or
  trough
• Wavelength distance wave travels in 1 cycle
  represented by lambda (l)

15
Longitudinal Waves

• Has wave motion particle motion in same
  direction (parallel)
• Compression compressed region of wave similar
  to crest high density region
• Rarefaction stretched region of wave similar
  to trough low density region

16
Speed of a Wave

• Source of all wave motion is vibrating obj
• Frequency of wave frequency of source
• Frequency of wave passing in a given amt of
  time
• Period time required for 1 complete vibration
  (time for 1 wavelength)

17
Speed of a Wave

• v fl
• Speed is constant for a medium
• Speed changes only when wave changes mediums or
  properties of the medium change
• Even if f is changed, l will change to make
  product (v) the same

18
Waves Transfer Energy

• Medium remains in basically same location
• Energy is transferred by wave from 1 place to
  another
• Matter vibrates transfers energy matter does
  not change location
• Amplitude energy are directly related
• Energy transfer amplitude2

19
Wave Interactions

• Waves are disturbances of matter (not matter) so
  they can occupy the same space
• Superposition combination of 2 overlapping
  waves
• Figure 15, page 389

20
Interference

• Displacements in the same direction produce
  constructive interference
• Waves maintain their original properties after
  interference
• Resultant wave is the sum of the amplitude
• Add amplitudes together to get larger waves
• See Figure 16, page 390

21
Interference

• Displacements in opposite directions produce
  destructive interference
• Occurs when crests troughs meet
• Resultant wave is difference between pulses
• Resultant has smaller amplitude
• If 2 waves have equal amplitude, difference is
  zero it is complete destructive interference

22
Interference

• Interference patterns are for both transverse
  longitudinal waves

23
Standing Waves

• Standing wave resultant wave pattern that
  appears to be stationary composed of alternating
  constructive/destructive interference
• Nodes pts on standing waves where there is
  complete destructive interference
• Antinodes constructive interference