Vibration and Waves

1
Vibration and Waves

• Chapter 11

2
Simple Harmonic Motion

• Section 1

3
Hookes Law

• A repeated motion such as a child on a swing is
  called periodic motion.
• Other examples would be a wrecking ball or a
  pendulum of a clock.
• In each case periodic motion is back and forth
  over the same path.

4
Hookes Law

• One type of periodic motion is the motion of a
  mass attached to a spring.
• The direction of the force acting on the mass
  (Felastic) is always opposite the direction of
  the masss displacement from equilibrium (x
  0).

5
Hookes Law

• At equilibrium
• The spring force and the masss acceleration
  become zero.
• The speed reaches a maximum.
• At maximum displacement
• The spring force and the masss acceleration
  reach a maximum.
• The speed becomes zero.

6
Hookes Law

• Measurements show that the spring force, or
  restoring force, is directly proportional to the
  displacement of the mass.
• This relationship is known as Hookes Law
• Felastic kx
• spring force (spring constant ? displacement)
• The quantity k is a positive constant called the
  spring constant.

7
Hookes Law

• In Hookes Law the negative sign in the equation
  signifies that the direction of the spring force
  is always opposite the direction of the masss
  displacement from equilibrium.
• k is the spring constant and measures the
  stiffness of the spring.
• The spring constant is always a positive value.

8
Example

• If a mass of 0.55 kg attached to a vertical
  spring stretches the spring 2.0 cm from its
  original equilibrium position, what is the spring
  constant?

9

• Fnet 0 Felastic Fg
• Felastic kx
• Fg mg
• kx mg 0

10
Solution

• Rearrange the equation

11
Your Turn I

• A load of 45 N attached to a spring that is
  hanging vertically stretches the spring 0.14 m.
  What is the spring constant?
• A slingshot consists of a light leather cup
  attached between two rubber bands. If it takes a
  force of 32 N to stretch the bands 1.2 cm, what
  is the equivalent spring constant of the two
  rubber bands?
• How much force is required to pull a spring 3.0
  cm from its equilibrium position if the spring
  constant is 2.7 x 103 N/m?

12
Simple Harmonic Motion

• The motion of a vibrating mass-spring system is
  an example of simple harmonic motion.
• Simple harmonic motion describes any periodic
  motion that is the result of a restoring force
  that is proportional to displacement.
• Because simple harmonic motion involves a
  restoring force, every simple harmonic motion is
  a back-and-forth motion over the same path.

13
The Simple Pendulum

• A simple pendulum consists of a mass called a
  bob, which is attached to a fixed string.
• At any displacement from equilibrium, the weight
  of the bob (Fg) can be resolved into two
  components.
• The x component (Fg,x Fg sin q) is the only
  force acting on the bob in the direction of its
  motion and thus is the restoring force.

The forces acting on the bob at any point are the
force exerted by the string and the
gravitational force.
14
The Simple Pendulum

• The magnitude of the restoring force (Fg,x Fg
  sin q) is proportional to sin q.
• When the maximum angle of displacement q is
  relatively small (lt15), sin q is approximately
  equal to q in radians.
• As a result, the restoring force is very nearly
  proportional to the displacement.
• Thus, the pendulums motion is an excellent
  approximation of simple harmonic motion.

15
Energy in Simple Harmonic Motion

• Potential Energy
• Kinetic Energy
• Mechanical Energy

Energy
Displacement
16
Simple Harmonic Motion
17
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• Honors 1-4

18
Amplitude, Period, and Frequency in SHM

• In SHM, the maximum displacement from equilibrium
  is defined as the amplitude of the vibration.
• A pendulums amplitude can be measured by the
  angle between the pendulums equilibrium position
  and its maximum displacement.
• For a mass-spring system, the amplitude is the
  maximum amount the spring is stretched or
  compressed from its equilibrium position.
• The SI units of amplitude are the radian (rad)
  and the meter (m).

19
Amplitude, Period, and Frequency in SHM

• The period (T) is the time that it takes a
  complete cycle to occur.
• The SI unit of period is seconds (s).
• The frequency (f) is the number of cycles or
  vibrations per unit of time.
• The SI unit of frequency is hertz (Hz).
• Hz s1

20
Amplitude, Period, and Frequency in SHM

• Period and frequency are inversely related
• Thus, any time you have a value for period or
  frequency, you can calculate the other value.

21
Measures of Simple Harmonic Motion
22
Period of a Simple Pendulum in SHM

• The period of a simple pendulum depends on the
  length and on the free-fall acceleration.
• The period does not depend on the mass of the bob
  or on the amplitude (for small angles).

23
Example

• You need to know the height of a tower, but
  darkness obscures the ceiling. You note that a
  pendulum extending from the ceiling almost
  touches the floor and that its period is 12 s.
  How tall is the tower?

24
Solution

• Choose the equation and rearrange for the length.
• L36 m

25
Your Turn II

• If the period of the pendulum in the example
  problem were 24 s, how tall would the building
  be?
• You are designing a pendulum clock to have a
  period of 1.0 s. How long should the pendulum
  be?
• A trapeze artist swings in a simple harmonic
  motion with a period of 3.8 s. Calculate the
  length of the cables supporting the trapeze.
• Calculate the period and frequency of a 3.500 m
  long pendulum at the following locations
• North Pole ag 9.832 m/s2
• Chicago - ag 9.803 m/s2
• Jakarta Indonesia - ag 9.782 m/s2

26
Period of a Mass-Spring System in SHM

• The period of an ideal mass-spring system depends
  on the mass and on the spring constant.
• The period does not depend on the amplitude.
• This equation applies only for systems in which
  the spring obeys Hookes law.

27
Example

• The body of a 1275 kg car is supported on a frame
  of four springs. Two people riding in the car
  have a combined mass of 153 kg. When driven over
  a pothole in the road, the frame vibrates with a
  period of 0.840 s. For the first few seconds,
  the vibration acts like that of simple harmonic
  motion. Find the spring constant of a single
  spring.

28
Solution

• Choose the equation and rearrange to solve for k.
• k 2.00 x 104 N/m

29
Your Turn III

• A mass of 0.30 kg is attached to a spring and is
  set into vibration with a period of 0.24 s. What
  is the spring constant of the spring?
• 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?
• A spring with a spring constant of 30.0 N/m is
  attached to different masses, and the system is
  set into motion. Find the period and frequency
  of vibration for masses with the following
  magnitudes
• 2.3 kg
• 15 g
• 1.9 kg

30
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31
Properties of Waves

• Section 3

32
Wave Motion

• Picture this . . .
• You throw a pebble into a pond.
• The disturbance generates water waves that travel
  away from the disturbance.
• If there was a leaf near the disturbance you
  would see it move up and down and back and forth,
  but the net displacement would be ZERO!

33
Wave Motion

• A wave is the motion of a disturbance.
• A medium is a physical environment through which
  a disturbance can travel. For example, water is
  the medium for ripple waves in a pond.
• Waves that require a medium through which to
  travel are called mechanical waves. Water waves
  and sound waves are mechanical waves.
• Electromagnetic waves such as visible light,
  radio waves, X-rays and microwaves do not require
  a medium.

34
Wave Types

• A wave that consists of a single traveling pulse
  is called a pulse wave.
• Whenever the source of a waves motion is a
  periodic motion, such as the motion of your hand
  moving up and down repeatedly, a periodic wave is
  produced.
• A wave whose source vibrates with simple harmonic
  motion is called a sine wave. Thus, a sine wave
  is a special case of a periodic wave in which the
  periodic motion is simple harmonic.

35
Relationship Between SHM and Wave Motion

• As the sine wave created by this vibrating blade
  travels to the right, a single point on the
  string vibrates up and down with simple harmonic
  motion.

36
Wave Types

• A transverse wave is a wave whose particles
  vibrate perpendicularly to the direction of the
  wave motion.
• The crest is the highest point above the
  equilibrium position, and the trough is the
  lowest point below the equilibrium position.
• The wavelength (l) is the distance between two
  adjacent similar points of a wave.

37
Wave Types

• Longitudinal waves are when the particles of the
  medium vibrate parallel to the direction of the
  wave motion.
• Sound waves are longitudinal waves.

38
Period, Frequency, and Wave Speed

• The frequency of a wave describes the number of
  waves that pass a given point in a unit of time.
• The period of a wave describes the time it takes
  for a complete wavelength to pass a given point.
• The relationship between period and frequency in
  SHM holds true for waves as well the period of a
  wave is inversely related to its frequency.

39
Period, Frequency, and Wave Speed
40
Period, Frequency, and Wave Speed

• The speed of a mechanical wave is constant for
  any given medium.
• The speed of a wave is given by the following
  equation
• v fl
• wave speed frequency ? wavelength
• This equation applies to both mechanical and
  electromagnetic waves.

41
Waves and Energy Transfer

• Waves transfer energy by the vibration of matter.
• Waves are often able to transport energy
  efficiently.
• The rate at which a wave transfers energy depends
  on the amplitude.
• The greater the amplitude, the more energy a wave
  carries in a given time interval.
• For a mechanical wave, the energy transferred is
  proportional to the square of the waves
  amplitude.
• The amplitude of a wave gradually diminishes over
  time as its energy is dissipated.

42
Example

• A piano string tuned to middle C vibrates with a
  frequency of 262 Hz. Assuming the speed of sound
  in air is 343 m/s, find the wavelength of the
  sound waves produced by the string.

43
Solution

• Choose your Equation
• Rearrange the equation

44
Solution

• Plug and Chug
• 343 m/s / 262 Hz
• 1.31 m

45
Your Turn IV

• A piano emits frequencies that range from a low
  of about 28 Hz to a high of about 4200 Hz. Find
  the range of wavelengths in air attained by the
  piano when the speed of sound in air is 340 m/s.
• The speed of electromagnetic waves in empty space
  is 3.00 x 108 m/s. Calculate the wavelength
  emitted at the following frequencies
• Radio waves at 88.0 MHz
• Visible light at 6.0 x 108 MHz
• X rays at 3.0 x 1012 MHz
• A tuning fork produces a sound with a frequency
  of 256 Hz and a wavelength in air of 1.35 m.
• What value does this give for the speed of sound
  in air?
• What would the wavelength be of this same sound
  in water in which sound travels at 1500 m/s?

46
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47
Wave Interactions

• Section 4

48
Wave Interference

• Two different material objects can never occupy
  the same space at the same time.
• Because mechanical waves are not matter but
  rather are displacements of matter, two waves can
  occupy the same space at the same time.
• The combination of two overlapping waves is
  called superposition.

49
Wave Interference

• In constructive interference, individual
  displacements on the same side of the equilibrium
  position are added together to form the resultant
  wave.

50
Wave Interference

• In destructive interference, individual
  displacements on opposite sides of the
  equilibrium position are added together to form
  the resultant wave.

51
Reflection

• What happens to the motion of a wave when it
  reaches a boundary?
• At a free boundary, waves are reflected.
• At a fixed boundary, waves are reflected and
  inverted.

Free boundary Fixed boundary
52
Standing Waves

• A standing wave is a wave pattern that results
  when two waves of the same frequency, wavelength,
  and amplitude travel in opposite directions and
  interfere.
• Standing waves have nodes and antinodes.
• A node is a point in a standing wave that
  maintains zero displacement.
• An antinode is a point in a standing wave,
  halfway between two nodes, at which the largest
  displacement occurs.

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Standing Waves

• Only certain wavelengths produce standing wave
  patterns.
• The ends of the string must be nodes because
  these points cannot vibrate.
• A standing wave can be produced for any
  wavelength that allows both ends to be nodes.
• In the diagram, possible wavelengths include 2L
  (b), L (c), and 2/3L (d).

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