Vibrations and Waves

1
Vibrations and Waves

• Chapter 12

2
Periodic Motion

• A repeated motion is called periodic motion
• What are some examples of periodic motion?
• The motion of Earth orbiting the sun
• A child swinging on a swing
• Pendulum of a grandfather clock

3
Simple Harmonic Motion

• Simple harmonic motion is a form of periodic
  motion
• The conditions for simple harmonic motion are as
  follows
• The object oscillates about an equilibrium
  position
• The motion involves a restoring force that is
  proportional to the displacement from equilibrium
• The motion is back and forth over the same path

4
Earths Orbit

• Is the motion of the Earth orbiting the sun
  simple harmonic?
• NO
• Why not?
• The Earth does not orbit about an equilibrium
  position

5
p. 438 of your book

• The spring is stretched away from the
  equilibrium position
• Since the spring is being stretched toward the
  right, the springs restoring force pulls to the
  left so the acceleration is also to the left

6
p. 438 of your book

• When the spring is unstretched the force and
  acceleration are zero, but the velocity is
  maximum

7
p.438 of your book

• The spring is stretched away from the equilibrium
  position
• Since the spring is being stretched toward the
  left, the springs restoring force pulls to the
  right so the acceleration is also to the right

8
Damping

• In the real world, friction eventually causes the
  mass-spring system to stop moving
• This effect is called damping

9
Mass-Spring Demo

• http//phet.colorado.edu/simulations/sims.php?sim
  Masses_and_Springs
• I suggest you play around with this demoit might
  be really helpful!

10
Hookes Law

• The spring force always pushes or pulls the mass
  back toward its original equilibrium position
• Measurements show that the restoring force is
  directly proportional to the displacement of the
  mass

11
Hookes Law

• Felastic Spring force
• k is the spring constant
• x is the displacement from equilibrium
• The negative sign shows that the direction of F
  is always opposite the mass displacement

12
Flashback

• Anybody remember where weve seen the spring
  constant (k) before?
• PEelastic ½kx2
• A stretched or compressed spring has elastic
  potential energy!!

13
Spring Constant

• The value of the spring constant is a measure of
  the stiffness of the spring
• The bigger k is, the greater force needed to
  stretch or compress the spring
• The units of k are N/m (Newtons/meter)

14
Sample Problem p.441 2

• A load of 45 N attached to a spring that is
  hanging vertically stretches the spring 0.14 m.
  What is the spring constant?

15
Solving the Problem

• Why do I make x negative?
• Because the displacement is down

16
Follow Up Question

• What is the elastic potential energy stored in
  the spring when it is stretched 0.14 m?

17
The simple pendulum

• The simple pendulum is a mass attached to a
  string
• The motion is simple harmonic
• because the restoring force is proportional to
  the displacement and because the mass oscillates
  about an equilibrium position

18
Simple Pendulum

• The restoring force is a component of the mass
  weight
• As the displacement increases, the gravitational
  potential energy increases

19
Simple Pendulum Activity

• http//phet.colorado.edu/simulations/sims.php?sim
  Pendulum_Lab
• You should also play around with this activity to
  help your understanding

20
Comparison between pendulum and mass-spring
system (p. 445)
21
Measuring Simple Harmonic Motion (p. 447)
22
Amplitude of SHM

• Amplitude is the maximum displacement from
  equilibrium
• The more energy the system has, the higher the
  amplitude will be

23
Period of a pendulum

• T period
• L length of string
• g 9.81 m/s2

24
Period of the Pendulum

• The period of a pendulum only depends on the
  length of the string and the acceleration due to
  gravity
• In other words, changing the mass of the pendulum
  has no effect on its period!!

25
Sample Problem p. 449 2

• You are designing a pendulum clock to have a
  period of 1.0 s. How long should the pendulum be?

26
Solving the Problem
27
Period of a mass-spring system

• T period
• m mass
• k spring constant

28
Sample Problem p. 451 2

• When a mass of 25 g is attached to a certain
  spring, it makes 20 complete vibrations in 4.0 s.
  What is the spring constant of the spring?

29
What information do we have?

• M .025 kg
• The mass makes 20 complete vibrations in 4.0s
• That means it makes 5 vibrations per second
• So f 5 Hz
• T 1/5 0.2 seconds

30
Solve the problem
31
Day 2 Properties of Waves

• A wave is the motion of a disturbance
• Waves transfer energy by transferring the motion
  of matter instead of transferring matter itself
• A medium is the material through which a
  disturbance travels
• What are some examples of mediums?
• Water
• Air

32
Two kinds of Waves

• Mechanical Waves require a material medium
• i.e. Sound waves
• Electromagnetic Waves do not require a material
  medium
• i.e. x-rays, gamma rays, etc

33
Pulse Wave vs Periodic Wave

• A pulse wave is a single, non periodic
  disturbance
• A periodic wave is produced by periodic motion
• Together, single pulses form a periodic wave

34
Transverse Waves

• Transverse Wave The particles move perpendicular
  to the waves motion

Particles move in y direction
Wave moves in X direction
35
Longitudinal (Compressional) Wave

• Longitudinal (Compressional) Waves Particles
  move in same direction as wave motion (Like a
  Slinky)

36
Longitudinal (Compressional) Wave
Crests Regions of High Density because The coils
are compressed
Troughs Areas of Low Density because The coils
are stretched
37
Wave Speed

• The speed of a wave is the product of its
  frequency times its wavelength
• f is frequency (Hz)
• ? (lambda) Is wavelength (m)

38
Sample Problem p.457 4

• A tuning fork produces a sound with a frequency
  of 256 Hz and a wavelength in air of 1.35 m
• a. What value does this give for the speed of
  sound in air?
• b. What would be the wavelength of the wave
  produced b this tuning fork in water in which
  sound travels at 1500 m/s?

39
Part a

• Given
• f 256 Hz
• ? 1.35 m
• v ?

40
Part b

• Given
• f 256 Hz
• v 1500 m/s
• ? ?

41
Wave Interference

• Since waves are not matter, they can occupy the
  same space at the same time
• The combination of two overlapping waves is
  called superposition

42
The Superposition Principle

• The superposition principle When two or more
  waves occupy the same space at the same time, the
  resultant wave is the vector sum of the
  individual waves

43
Constructive Interference (p.460)

• When two waves are traveling in the same
  direction, constructive interference occurs and
  the resultant wave is larger than the original
  waves

44
Destructive Interference

• When two waves are traveling on opposite sides of
  equilibrium, destructive interference occurs and
  the resultant wave is smaller than the two
  original waves

45
Reflection

• When the motion of a wave reaches a boundary, its
  motion is changed
• There are two types of boundaries
• Fixed Boundary
• Free Boundary

46
Free Boundaries

• A free boundary is able to move with the waves
  motion
• At a free boundary, the wave is reflected

47
Fixed Boundaries

• A fixed boundary does not move with the waves
  motion (pp. 462 for more explanation)
• Consequently, the wave is reflected and inverted

48
Standing Waves

• When two waves with the same properties
  (amplitude, frequency, etc) travel in opposite
  directions and interfere, they create a standing
  wave.

49
Standing Waves

• Standing waves have nodes and antinodes
• Nodes The points where the two waves cancel
• Antinodes The places where the largest amplitude
  occurs
• There is always one more node than antinode

A
50
Sample Problem p.465 2

• A string is rigidly attached to a post at one
  end. Several pulses of amplitude 0.15 m sent down
  the string are reflected at the post and travel
  back down the string without a loss of amplitude.
  What is the amplitude at a point on the string
  where the maximum displacement points of two
  pulses cross? What type of interference is this?

51
Solving the Problem

• What type of boundary is involved here?
• Fixed
• So that means the pulse will be reflected and
  inverted
• What happens when two pulses meet and one is
  inverted?
• Destructive interference
• The resultant amplitude is 0.0 m

52
Helpful Simulations

• Mass-Spring system http//phet.colorado.edu/simul
  ations/sims.php?simMasses_and_Springs
• Pendulum http//phet.colorado.edu/simulations/sim
  s.php?simPendulum_Lab
• Wave on a string system http//phet.colorado.edu/
  simulations/sims.php?simWave_on_a_String
• http//www.walter-fendt.de/ph14e/stwaverefl.htm