Glencoe Physics 2006

1
Chapter 14

• Glencoe Physics 2006
• Vibrations and Waves

2

• A. Periodic Motion
• 1. motions that repeat in a cyclical fashions
  back and forth. Study two types- uniform
  circular motion and simple harmonic motion
• 2. if the force restoring object to equilibrium
  position is directly proportional to the
  displacement of the object it is said to be in
  simple harmonic motion (SHM)
• 3. SHM object will have one position at which the
  net force is zero called the equilibrium
  position

3

• a. two properties describe SHM amplitude and
  period
• b. Amplitude, A
• (1) The amplitude is the maximum position of the
  object relative to the equilibrium position
• (2) In the absence of friction, an object in
  simple harmonic motion will oscillate between the
  positions x A

4

• c. Period, T
• (1) the time that it takes for the object to
  complete one complete cycle of motion
• (2) From xA to x -A and back to x A
• 4. Mass on a Spring Example of SHM
• a. Hookes Law
• (1) Fs - k x

5

• Fs is the spring force
• k is the spring constant
• (2) k is a measure of the stiffness of the spring
• (a) A large k indicates a stiff spring and a
  small k indicates a soft spring
• (3) x is the displacement of the object from its
  equilibrium position
• x 0 at the equilibrium position

6

• (4) The negative sign indicates that the force is
  always directed opposite to the displacement
• b. Hookes Law Force
• (1) The force always acts toward the equilibrium
  position
• (2) It is called the restoring force
• (3) The direction of the restoring force is such
  that the object is being either pushed or pulled
  toward the equilibrium position

7
5. Simple Harmonic Motion Requirement

• a. Motion that occurs when the net force along
  the direction of motion obeys Hookes Law
• The force is proportional to the displacement and
  always directed toward the equilibrium position
• b. The motion of a spring mass system is an
  example of Simple Harmonic Motion

8

• c. Not all periodic motion over the same path can
  be considered Simple Harmonic motion
• d. To be Simple Harmonic motion, the force needs
  to obey Hookes Law

9
6. Elastic Potential Energy

• a. A compressed spring has potential energy
• (1) The compressed spring, when allowed to
  expand, can apply a force to an object
• (2) The potential energy of the spring can be
  transformed into kinetic energy of the object

10

• b. The energy stored in a stretched or compressed
  spring or other elastic material is called
  elastic potential energy
• PEs ½kx2
• c. The energy is stored only when the spring is
  stretched or compressed
• d. Elastic potential energy can be added to the
  statements of Conservation of Energy and
  Work-Energy

11
7. Simple Pendulum

• a. The simple pendulum is another example of
  simple harmonic motion
• b. The force is the component of the weight
  tangent to the path of motion
• Ft - m g sin ?

12

• c. In general, the motion of a pendulum is not
  simple harmonic
• d. However, for small angles, it becomes simple
  harmonic
• In general, angles lt 15 are small enough
• sin ? ?
• Ft - m g ?
• This force obeys Hookes Law

13
8. Period of Simple Pendulum

• a. This shows that the period is independent of
  the amplitude
• b. The period depends on the length of the
  pendulum and the acceleration of gravity at the
  location of the pendulum

14
9. Physical Pendulum

• a. A physical pendulum can be made from an object
  of any shape
• b. The center of mass oscillates along a circular
  arc

15

• 10. Resonance
• a. occurs when small forces are applied at
  regular intervals to a oscillating object and the
  amplitude of the vibration increases.
• (1) time interval is equal to period
• (2) pumping a swing, jumping on a trampoline
• b. special form of SHM

16

• B. Wave Properties
• 1. Energy can be transferred between two points
  by either particles or by waves.
• a. Waves carry energy and momentum without the
  transfer of matter - true for all types of waves.
• b. Difference between throwing a ball at a target
  (it gains KE) and tying a rope to the target and
  shaking it.
• 2. A wave is the motion of a continuous
  disturbance.

17
3. Types of Waves

• a. Mechanical Waves - need a medium through
  which they can travel.
• (1) Examples - water waves, sound waves, spring
  waves
• (2) Material in which the disturbance is moving
  is called the medium and can be readily seen.

18
b. Electromagnetic Waves

• (1) Require no medium through which to travel
• (2) Example - light, radio, x-rays
• (3) E-M waves cannot be readily observed

19
c. Classification of Mechanical Waves - based on
way in which matter is displaced

• (1) Transverse Wave - particles in the medium
  vibrate perpendicular to the direction of the
  waves motion itself. Sometimes called traveling
  waves, as in a bump passing down a string or
  electromagnetic waves.

particle motion
direction of wave motion
20
(2) Longitudinal Waves - particles move parallel
to the direction of the wave.

• (a) Sound is an example, with compressions and
  rarefactions

rarefaction
compression
21
(3) Surface Wave - characteristics of both
transverse and longitudinal waves

• (a) Particles in medium move both horizontally
  and vertically.
• (b) Water waves are an example.
• (c) Although the particles in the medium move in
  response to a passing wave, they do not move
  along with the wave.

22
(4) Pulse - a single disturbance passing through
a medium.

• Ex

(5) (Periodic ) Wave - produced by a
periodic disturbance in which a series of pulses
at regular intervals pass through a medium.
23
d. Wave Characteristics

• (1) Wavelength (?) - the horizontal distance
  between corresponding points on consecutive waves.

?
crest
v
trough
(a) Tops are crests, bottoms are troughs (b)
wavelength measured in meters
24

• (2) Frequency (f) - number of wavelengths that
  pass a given point per second
• (a) Measured in hertz (hz)
• (b) one hertz 1 wave per second
• (3) Period (T) - time required for one complete
  wave to pass a given point
• (a) Measured in seconds
• (b) T 1 / f

25

• (4) Velocity (v) v f ? ? / T
• (a) velocity is in meters/ sec
• (b) for a given medium the speed of the wave is
  fixed.
• (5) Amplitude (A) - maximum displacement from the
  rest or equilibrium position
• (a) Energy content of a mechanical wave
  characterized by its amplitude.
• (b) Greater the amplitude, the more energy
  transferred and the more work done by the wave.

26

• C. Wave Behavior
• 1. Speed in a Medium
• a. mechanical wave has a constant speed in a
  given medium,
• (1) depends on the properties of the medium
• (2) change properties then change speed
• b. if frequency changes, wavelength changes.
• c. amplitude has no affect on speed

27

• 2. Behavior at Boundaries
• a. Rules apply to both transverse and
  longitudinal waves, but easier to see in
  transverse waves.
• b. Wave hits a boundary - part is reflected and
  remainder is transmitted into the new medium.
• (1) wave that strikes boundary is called the
  incident wave

28

• (2) the returning wave is called the reflected
  wave and its orientation depends on the
  differences between the two media.
• (3) If there is little difference between the two
  media, then only a little energy is reflected -
  the majority is transmitted into the new medium.
  Example is a spring and string connection.
• (4) Inversion - when wave is reflected at a more
  rigid (dense) medium.

29

• (a) Why? Newtons Third Law - pulse exerts a
  force upward on the wall, wall exerts a force
  downward, causing inversion.
• (b) Less dense medium to a more dense medium, get
  inversion.
• (5) If the spring is free to slide at the end
  (not rigid) then the wave is reflected erect.
  More dense to less dense medium, reflected wave
  is upright.

30

• c. Transmitted Waves
• (1) Speed of wave determined by the medium. Wave
  in a new medium generated by the wave in the old
  medium, thus has the same frequency.
• (2) With change in speed, however, you have a
  change in wavelength.

31

• v f ? lower velocity, shorter ?
  higher velocity, longer ?
• 3. Principle of Superposition
• a. Two or more waves, each wave affects the
  medium independently. Similar to vectors.
• b. At the point where the two or more waves
  meet, displacement of the medium is the sum of
  the displacements of the individual waves. Add
  the amplitudes.

32
4. Interference - effect of two of more waves or
pulses on each other when traveling through a
medium

• a. Constructive Interference - add amplitudes
  together to produce a greater amplitude. Waves
  themselves pass through one another and are
  unaffected.
• b. Destructive Interference - waves combine to
  produce a pulse with smaller amplitude than
  either original.

33

• c. After two pulses pass through one another
  they return to their original shape, they are not
  changed by passing.
• d. If the waves have the same wavelengths but
  different amplitudes, the vector sum of the
  displacements gives the amplitude at the point of
  passing.

34
5. Nodes - consider two pulses with equal but
opposite amplitudes and the same shape.

• a. nodes are points at which medium is
  undisturbed.

?
v
node
node
b. medium does not undergo displacement at a
node
35
c. nodes are the result of destructive
interference

• 6. Antinodes
• a. points of maximum displacement
• b. result of constructive interference

v
antinode
36
7. Standing Waves

• a. periodic wave combines with its reflected
  wave so that the pattern appears to stand still.
• b. result of resonance - which is an additive
  affect of the amplitude of the wave.

37
c . Examples of standing waves

string oscillating against a building
?
Barrier
oscillator
50 centimeters
What you see is a sine wave reflected back on
itself . Notice the nodes and antinodes.
38

• 8. The Law of Reflection - the angle of
  reflection is equal to the angle of incidence
• a. normal - the line that is perpendicular to the
  surface at the point of reflection
• b. angles are always measured from the incident
  ray to the normal and from the reflected wave to
  the normal

normal
i
reflected wave
r
incident ray
surface
39

• 9. Refraction - change in wave direction at the
  boundary between two media
• a. Same frequency, since waves in the second
  medium are caused by waves in the first medium,
  but different velocities. Key is velocity
  changes with medium.
• b. Since v f ? , as ? gets smaller, velocity
  must decrease and waves bend.

40

• 10. Diffraction
• a. Diffraction - the bending of a wave around
  obstacles placed in its path. Based on the
  concept that every point on a wavefront acts as a
  point source.

41

• In the standing wave shown, what is its
  amplitude? What is its wavelength? How many nodes
  are there?

2.5 meters
20 centimeters
42

• The amplitude of the wave is 10 centimeters the
  wavelength is 1 meter and, there are 6 nodes.