Vibrations and Waves

1
Chapter 12

• Vibrations and Waves

2
Chapter 12 section 1

• Objectives
• Identify the conditions of simple harmonic
  motion.
• Explain how force, velocity, and acceleration
  change as an object vibrates with simple harmonic
  motion.
• Calculate the spring force using Hookes law.

3
Periodic Motion

• repeated motion
• back and forth over the same path
• ex. mass attached to a spring, pendulum, swinging
• at equilibrium (starting point), the force is
  0
• at equilibrium, the acceleration is 0
• F and mass decrease as spring moves toward
  equilibrium
• velocity increases as spring moves toward
  equilibrium
• at equilibrium velocity reaches its maximum

4
Periodic Motion

• at maximum displacement, force and acceleration
  are at their maximum
• at maximum displacement, velocity 0
• friction causes the spring to come to rest
  (dampening)

5
Demo.

• periodic motion
• Spring
• pendulum

6
Restoring Force

• Force that pushes or pulls the mass back toward
  its original equilibrium position
• Directly proportional to the displacement of the
  mass

7
Simple Harmonic Motion

• motion that is back-and-force motion over the
  same path
• involves a restoring force

8
Hookes Law

• Felastic - kx
• k spring constant
• Stiffness of spring
• Unit is N/m
• Negative b/c direction of force is always
  opposite the direction of mass displacement

9
Demo.

• Hookes Law with two different springs

10
Examples

• A 76 N crate is attached to a spring w/a spring
  constant of 450 N/m. How much displacement is
  caused by the weight of the crate?
• F 76 N k 450 N/m x ?
• F - kx 76 -(450)(x)
• x -0.17 m

11
Energy in Springs

• a spring stretched has all PE
• At rest
• when spring is released, PE is converted to KE
• all E is then KE
• initial PE must be equal to final KE

12
Review and Assignment

• Identify the conditions of simple harmonic
  motion.
• Explain how force, velocity, and acceleration
  change as an object vibrates with simple harmonic
  motion.
• Calculate the spring force using Hookes law.
• Page 441 1 4

13
Chapter 12 section 2

• Objectives
• Identify the amplitude of vibration.
• Recognize the relationship between period and
  frequency.
• Calculate the period and frequency of an object
  vibrating with simple harmonic motion.

14
Amplitude

• maximum displacement from equilibrium
• for a pendulum, it is measured by the angle btwn
  equilibrium and maximum displacement
• for mass-spring, it is measured by maximum
  amount that spring is stretched or compressed
  from equilibrium position

15
Period (T)

• time to complete one whole cycle
• One side to the other and back to starting point
• Unit is the second
• For a pendulum, d/o string length and free fall
  acceleration
• When shorter, smaller arc to travel through
• T 2pvL/g

16
Period (T)

• for a mass-spring system, d/o mass and spring
  constant
• increase k, then increase F to compress or
  stretch spring
• increase m, then increase time of cycle
• T 2pvm/k

17
Frequency - ()

• of cycles in a unit time
• unit is cycles/sec or Hertz
• and T are inversely related
• 1/T and T 1/

18
Demo.

• pendulum string length

19
Review and Assignment

• Identify the amplitude of vibration.
• Recognize the relationship between period and
  frequency.
• Calculate the period and frequency of an object
  vibrating with simple harmonic motion.
• Page 449 1 4 and 451 1 - 5

20
Chapter 12 section 2 day 2

• Objectives
• Explain how simple harmonic motion principles are
  applied to pendulums.

21
Simple Pendulums

• consists of a mass called a bob, attached to a
  string
• disregard friction and air resistance
• forces acting on the bob include force of the
  string (y axis) and the bobs weight (x axis)
• bobs weight is the restoring force
• magnitude of restoring force d/o distance from
  starting position

22
Restoring Force

• is zero at equilibrium
• decreases as it moves toward equilibrium
  position
• at angles under 15 the restoring force is
  proportional to the displacement and motion is
  simple harmonic

23
Pendulum

• at maximum displacement, restoring force and
  acceleration are at maximum
• at maximum displacement, velocity 0
• at equilibrium, force and acceleration 0
• at equilibrium, velocity reaches a maximum

24
Energy and Pendulums

• PE is gravitational
• PE is 0 at the lowest point of the swing
• at maximum displacement, E is all Peg
• as it swings toward equilibrium it gains KE and
  loses PE
• At equilibrium, E is all KE

25
Quick Lab

• p. 444
• Car and pendulum

26
Review and Assignment

• Explain how simple harmonic motion principles are
  applied to pendulums.
• Page 445 1 - 4

27
Chapter 12 section 3

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

28
Wave

• motion of disturbance
• E is transferred from point to point

29
Medium

• medium through which a disturbance travels
• some waves need a medium to travel
• ex. sound
• medium does NOT travel w/wave, it returns to its
  original position
• ex. air, water, glass

30
Demo.

• spring and flag

31
Mechanical Waves

• waves that require a medium to travel through
• sound

32
Electromagnetic Waves

• do not need a medium to travel through
• Light, radio, microwaves, x-rays

33
Pulse Wave

• a wave w/a single traveling pulse

34
Periodic Wave

• wave whose source is a form of periodic motion
• Many pulse waves

35
Sine Waves

• describes particles that vibrate w/simple
  harmonic motion
• an example of a periodic wave w/simple harmonic
  motion

36
Transverse Waves

• wave whose particles vibrate perpendicularly to
  the direction of wave motion
• demo.
• slinky

37
Crest

• highest point above the equilibrium position

38
Trough

• lowest point below the equilibrium position

39
Wavelength

• distance btwn two similar points of the wave
• Crest to crest
• Trough to trough

40
Longitudinal Wave

• wave whose particles vibrate parallel to the
  direction of wave motion
• ex. stretching and compressing a spring
• ex. sound waves in air
• Can be described w/a sine curve
• demo.
• slinky

41
Velocity of Waves

• v ?
• is constant for any given medium
• is inversely proportional
• If one increases, the other decreases
• Speed only changes when moving from one medium to
  another
• demo.
• slinky

42
Wave Energy

• E is transferred while the water remains in the
  same place
• They do this by transferring the motion of
  matter, not the matter itself
• That is why they can transfer E efficiently
• Transfer of E depends on the amplitude of
  vibration
• If the amplitude is doubled the E increases by
  four
• E transfer is proportional to the square of the
  amplitude

43
Review and Assignment

• Distinguish local particle vibrations from
  overall wave motion.
• Differentiate between pulse waves and periodic
  waves.
• Interpret waveforms of transverse and
  longitudinal waves.
• Apply the relationship among wave speed,
  frequency, and wavelength to solve problems.
• Relate energy and amplitude.
• Page 457 1 4

44
Chapter 12 section 4

• Objectives
• Apply the superposition principle.
• Differentiate between constructive and
  destructive interference.
• Predict when a reflected wave will be inverted.
• Predict whether specific traveling waves will
  produce a standing wave.
• Identify nodes and antinodes of a standing wave.

45
Wave Interference

• two waves can occupy the same space at the same
  time, b/c they are not made of matter
• this is called superposition
• as the waves interact they form interference
  patterns

46
Constructive Interference

• individual displacements on the same side of
  equilibrium position are added together
• constructive wave

47
Superposition Principle

• when two or more waves travel through a medium
  the resultant wave is the sum of the
  displacements of the individual waves at each
  point
• Valid only for small amplitude waves
• superposition

48
Destructive Interference

• individual displacements on opposite sides of
  the equilibrium position are added together
• One is and one is
• If they are equal the resultant is zero
• Complete destructive interference
• destructive interference
• both type of interference work on transverse and
  longitudinal waves

49
Reflection

• waves bounce off of an object and reverse in
  direction
• w/a fixed end, waves are both reflected and
  inverted
• demo.
• Inverted wave

50
Standing Waves

• a wave pattern that results when two waves of
  the same frequency, wavelength, and amplitude
  travel in opposite directions and interfere
• Alternating regions of constructive and
  destructive interference
• Have nodes and antinodes

51
Nodes

• point in a standing wave that always undergoes
  complete destructive interference and is
  stationary
• No motion in the wave at these points

52
Antinodes

• point in a standing wave halfway btwn two nodes
  at which the largest amplitude occurs

53
Standing Waves

• only certain frequencies will produce standing
  wave patterns
• all points have the same frequency
• ends must be nodes and cannot vibrate
• standing waves can be produced for a frequency
  that allows both end to be nodes

54
Standing Wave Lengths

• 1. two ends stationary 1 loop
• 1 loop ½ of a wavelength
• So to have one full wave you need 2L
• L length of string
• 2. three nodes
• Both ends and the middle
• 2 loops 1 full wavelength
• L 1
• 3. four nodes
• Both ends and two in the middle
• 3 loops 1 and ½ waves
• Need 2/3 L

55
Demo.

• standing waves

56
Review and Assignment

• Apply the superposition principle.
• Differentiate between constructive and
  destructive interference.
• Predict when a reflected wave will be inverted.
• Predict whether specific traveling waves will
  produce a standing wave.
• Identify nodes and antinodes of a standing wave.
• Page 465 1 5