1. Vibration and Waves
AP Physics
Chapter 11

2. 11.1 Simple Harmonic Motion
Vibration and Waves
11.1 Simple Harmonic Motion

3. 11.1 Simple Harmonic Motion
Periodic motion – when an object vibrates over the same pathway, with each vibration taking the same amount of time
Equilibrium position – the
position of the mass when
no force is exerted on it
11.1

4. 11.1 Simple Harmonic Motion
If the spring is stretched from equilibrium, a force acts so the object is pushed back toward equilibrium
Restoring Force
Proportional to the
displacement (x)
Called Hooke’s Law
11.1

5. 11.1 Simple Harmonic Motion
Any vibrating system for which the restoring force is directly proportional to the negative of the displacement (F = -kx) exhibits Simple Harmonic Motion (SHM)

6. 11.1 Simple Harmonic Motion
Amplitude (A) – maximum distance from equilibrium
Period (T) – time for one
complete cycle
Frequency (f) – number
of vibrations per
second
Amplitude
11.1

7. 11.1 Simple Harmonic Motion
A vertical spring follows the same
pattern
The equilibrium positions is just
shifted by gravity
11.1

8. 11.2 Energy in a Simple Harmonic Oscillator
Vibration and Waves
11.2 Energy in a Simple Harmonic Oscillator

9. 11.2 Energy in a Simple Harmonic Oscillator
Review – energy of a spring
So the total mechanical energy of a spring (assuming no energy loss) is
At maximum amplitude then
So E is proportional to the square of the amplitude
11.2

10. 11.2 Energy in a Simple Harmonic Oscillator
Velocity as a function of position
Since the maximum velocity is when A=0
Factor the top equation for A2,
then combine with the bottom equation
Take square root
11.2

11. 11.3 The Period and Sinusoidal Nature of SHM
Vibration and Waves
11.3 The Period and Sinusoidal Nature of SHM

12. 11.3 The Period and Sinusoidal Nature of SHM
The Period of an object undergoing SHM is independent of the amplitude
Imagine an object traveling in a circular pathway
If we look at the motion in
just the x axis, the
motion is analogous
to SHM
11.3

13. 11.3 The Period and Sinusoidal Nature of SHM
As the ball moves the displacement in the x changes
The radius is the amplitude
The velocity is tangent to the
circle
Now looking at Components
If we put the angles into the
triangle, we can see
similar triangles
Vmax q V A q
11.3

14. 11.3 The Period and Sinusoidal Nature of SHM
So the opposite/hypotenuse is a constant
This can be rewritten
This is the same as the
equation for velocity of
an object in SHM
Vmax q V A q
11.3

15. 11.3 The Period and Sinusoidal Nature of SHM
The period would be the time for one complete revolution
The radius is the same as
the amplitude, and the
time for one revolution
is the period
11.3

16. 11.3 The Period and Sinusoidal Nature of SHM
Using the previous relationship between maximum velocity and amplitude
Substitute in the top equation
11.3

17. 11.3 The Period and Sinusoidal Nature of SHM
Position as a function of time
x displacement is
You don’t know this, but
Where 𝑓 is the frequency So A q
11.3

18. Vibration and Waves
11.4 The Simple Pendulum

19. Cord is massless (or very small) Mass is concentrated in small volume
11.4 The Simple Pendulum
Simple Pendulum – mass
suspended from a cord
Cord is massless (or very
small)
Mass is concentrated in small volume
11.4

20. Looking at a diagram of a pendulum Two forces act on the it Weight
11.4 The Simple Pendulum
Looking at a diagram of a pendulum
Two forces act on the it
Weight
Tension
The motion of the bob is at a
tangent to the arc T W
11.4

21. The displacement of the bob is given by x Using the triangle
11.4 The Simple Pendulum
The displacement of the bob is given by x
Using the triangle
For a complete circle (360o)
Then for our arc it would be q L Ly T W x
11.4

22. The component of force in the direction of motion is
11.4 The Simple Pendulum
The component of force in the direction of
motion is
But at small angle
And So q L Ly T
mgsinq W x
11.4

23. Usual standard is below 30o
11.4 The Simple Pendulum
Usual standard is below 30o
We can then take the equation and reason that 360o=2p rad (we didn’t study angular motion, so take my word for it)
11.4

24. We can now combine the equation for period
11.4 The Simple Pendulum
We can now combine the equation for period
Mass does not appear in this equation
The period is independent of mass
11.4

25. 11.5 Damped Harmonic Motion
Vibration and Waves
11.5 Damped Harmonic Motion

26. 11.5 Damped Harmonic Motion
The amplitude of a real oscillating object will decrease with time – called damping
Underdamped – takes several swing before coming to rest (above)
11.5

27. 11.5 Damped Harmonic Motion
Overdamped – takes a long time to reach equilibrium
Critical damping – equalibrium reached in the shortest time
11.5

28. 11.6 Forced Vibrations; Resonance
Vibration and Waves
11.6 Forced Vibrations; Resonance

29. 11.6 Forced Vibrations; Resonance
Natural Frequency – depends on the variables (m,k or L,g) of the object
Forced Vibrations –
caused by an
external force
11.6

30. 11.6 Forced Vibrations; Resonance
Resonant Frequency – the natural vibrating frequency of a system
Resonance – when the external frequency is near the natural frequency and damping is small
Tacoma Narrow Bridge
11.6

31. Vibration and Waves
11.7 Wave Motion

32. Mechanical Waves – travels through a medium
11.7 Wave Motion
Mechanical Waves – travels through a medium
The wave travels through the medium, but the medium undergoes simple harmonic motion
Wave motion
Particle motion
11.7

33. Waves transfer energy, not particles
11.7 Wave Motion
Waves transfer energy, not
particles
A single bump of a wave is called a pulse
A wave is formed when a force is applied to one end
Each successive particle is moved by the one next to it
Tsunami
11.7

34. perpenduclar to wave motion Wavelength (l) measured in meters
Parts of a wave
Transverse wave
– particle
motion
perpenduclar to wave motion
Wavelength (l) measured in meters
Frequency (f) measured in Hertz (Hz)
Wave Velocity (v) meters/second
11.7

35. Longitudinal (Compressional) Wave Particles move parallel to the
11.7 Wave Motion
Longitudinal (Compressional) Wave
Particles move
parallel to the
direction of wave motion
Rarefaction – where
particles are spread
out
Compression – particles
are close
11.7

36. Surface Waves – can travel along the boundary
11.7 Wave Motion
Earthquakes
S wave – Transverse
P wave – Longitudinal
Surface Waves – can travel along the boundary
Notice the circular motion of the particles
11.7

37. 11.9 Energy Transported by Waves
Vibration and Waves
11.9 Energy Transported by Waves

38. 11.9 Energy Transported by Waves
Energy for a particle undergoing simple harmonic motion is
Intensity (I) power across a unit area perpendicular to the
direction of energy
flow So
11.9

39. 11.11 Reflection and Transmission of Waves
Vibration and Waves
11.11 Reflection and Transmission of Waves

40. 11.11 Reflection and Transmission of Waves
When a wave comes to a
boundary (change in
medium) at least some of
the wave is reflected
The type of reflection depends
on if the boundary is fixed
(hard) - inverted
11.11

41. 11.11 Reflection and Transmission of Waves
When a wave comes to a
boundary (change in
medium) at least some of
the wave is reflected
Or movable (soft) – in phase
11.11

42. 11.11 Reflection and Transmission of Waves
For two or three dimensional we think in terms of wave fronts
A line drawn perpendicular to the wave front is called a ray
When the waves get far from their source and are nearly straight, they are called plane waves
11.11

43. 11.11 Reflection and Transmission of Waves
Law of Reflection – the angle of reflection equals the angle of incidence
Angles are always measured from
the normal
11.11

44. 11.12 Interference; Principle of Superposition
Vibration and Waves
11.12 Interference; Principle of Superposition

45. 11.12 Interference; Principle of Superposition
Interference – two waves pass through the same region of space at the same time
The waves pass through each other
Principle of Superposition – at the point where the waves meet the displacement of the medium is the algebraic sum of their separate displacements
11.12

46. 11.12 Interference; Principle of Superposition
Phase – relative position of the wave crests
If the two waves are “in phase”
Constructive interference
If the two waves are “out of phase”
Destructive Interference
11.12

47. 11.12 Interference; Principle of Superposition
For a wave (instead of a single phase)
Interference is
calculated by adding
amplitude
In real time this looks
like
11.12

48. 11.13 Standing Waves; Resonance
Vibration and Waves
11.13 Standing Waves; Resonance

49. 11.13 Standing Waves; Resonance
In a specific case of interference a standing wave is produced
The areas with complete constructive interference are called loops or antinodes (AN)
The areas with complete destructive interference are called nodes (N)
11.13

50. 11.13 Standing Waves; Resonance
Standing waves occur at the natural or resonant frequency of the medium
In this case, called the first harmonic, the wavelength is twice the length of the medium
The frequency of is called the fundamental frequency
11.13

51. 11.13 Standing Waves; Resonance
The second harmonic is the next standing wave formed
Then the third harmonic would be
11.13

52. 11.13 Standing Waves; Resonance
The basic form for the wavelength
of harmonics is
Each resonant frequency, is an integer multiple of the fundamental frequency
Overtone – all the frequencies above the fundamental
The first overtone is the second harmonic
Harmonic Applet
11.13