1. Physics Ch. 12 Vibrations and Waves
Any repeated motion over the same path is called a periodic motion
Ex: a child swinging on a swing, a pendulum on a grandfather clock, an acrobat swinging on a trapeze
Fig mass attached to a spring
The spring exerts a force on the mass when the spring is stretch or compressed.

2. At unstretched position, the spring is at equilibrium.
The force decreases as the mass approaches equilibrium and becomes zero at equilibrium
The mass’s acceleration becomes zero at equilibrium
The velocity reaches maximum at equilibrium.
The mass’s momentum causes the mass to overshoot the equilibrium and compress the spring

3. 12-1c: When the spring’s compression is equal to the stretched distance, the mass is at maximum displacement and the spring force and acceleration reach their maximum
Velocity at this point reaches zero.
The force works in opposite direction of the mass’s direction.

4. In an ideal situation, the mass would vibrate back and forth indefinitely
In the real world, friction reduces the motion and it eventually stops.
This effect is called damping.

5. Restoring force-the spring force pushing the mass back toward its original equilibrium position.
Restoring force directly proportional to the displacement of the mass. (Simple Harmonic Motion)

6. Hooke’s Law Felastic = -Kx
Spring force = -(spring constant X displacement)
K = spring constant (measure of the stiffness of the spring)

7. Sample Problem 12A
If a mass of 0.55 kg attached to a vertical spring stretches the spring 2.0 cm from its original equilibrium position, as shown in fig. 12-2, what is the spring constant?

8. Practice Problem
A 76 N crate is attached to a spring (k=450 N/m). How much displacement is caused by the weight of this crate?

9. Practice Problem
A spring of k=1962 N/m loses its elasticity if stretched more than 50.0 cm. What is the mass of the heaviest object the spring can support without being damaged?

10. Elastic Potential Energy
A stretched or compressed spring has stored elastic potential energy.
Ex: an archer that pulls the bowstring back has only elastic potential energy
When the bowstring is released, the elastic potential energy is converted to kinetic energy.
Because energy is conserved, the elastic potential energy is converted to kinetic energy of the arrow, bow, and bowstring.

11. The Simple Pendulum
A simple pendulum has a mass, called a bob, attached to a fixed string.
We assume that the mass of the bob is concentrated at a point and the mass of the string is negligible. We also disregard the effects of friction or air resistance.
For a physical pendulum, such as an acrobat, we will assume the same.

12. Restoring Force Which force on a pendulum acts as the restoring force?
The forces on the pendulum are the force of the string and the force of the bob’s weight. The force of the weight can be split into two components, x and y. The y component is opposite the force of the string and cancels. The net force on the bob is the x component of its weight.
This force then pushes or pulls the motion toward equilibrium, the restoring force.

13. The restoring force varies with the position to the equilibrium
Decreases as it approaches equilibrium and becomes zero at equilibrium
At small angles (<15°) the motion is simple harmonic

14. At maximum displacement, the restoring force and the acceleration are at maximum while the velocity is zero.
At equilibrium, the restoring force and acceleration become zero and velocity reaches a maximum
Table 12-1

15. Conservation of Energy
Spring-elastic potential energy
Pendulum-gravitational potential energy
At equilibrium, the gravitational potential energy is at zero, while the energy is solely kinetic energy
At maximum displacement, the kinetic energy turns entirely to gravitational potential energy

16. QuickLab pg. 444
Energy of a Pendulum

17. 12-2: Measuring Simple Harmonic Motion
Amplitude-maximum displacement from equilibrium (Can be measured by the angle or by the amount the spring is stretched or compressed)
Period (T)-the time it takes a complete cycle of motion; the pendulum ends of where it started
Frequency (f)-the number of cycles per second or the SI unit is Hertz (Hz)

18. These two units are inversely related:
F = or T = 1
T f

19. Very Quick Lab-4 Period and Frequency
Attach the pendulum bob to the string and suspend the string from the ring stand. Set the pendulum in motion. Have a student record the time required to complete 20 oscillations. Then, have another student record how many times the pendulum bob returns to the same place each second.

20. Have students use the first measurement to find the pendulum’s period
Have students use the first measurement to find the pendulum’s period. Use T=number of seconds/20 Ask the students what the second measurement indicates. Compare the values.

21. Period of a pendulum depends on Length of pendulum
If you have two pendulums with two different small amplitudes, the periods would still be the same. Thus, the period does not depend on the amplitude (<15º).
However, if the length of the pendulum string were changed or the free-fall acceleration, then the period would change.

22. Period of a simple pendulum in simple harmonic motion
T = 2πL g
Period=2π X square root of (length/free-fall acceleration)

23. Did you know?
Galileo is credited as the first person to notice that the motion of a pendulum depends on its length and is independent of its amplitude (for small angles). He supposedly observed this while attending church services at a cathedral in Pisa. The pendulum he studied was a swinging chandelier that was set in motion when someone bumped it while lighting the candles. Galileo is said to have measured its frequency, and hence its period, by timing the swings with his pulse.

24. Very Quick Lab-5-Relationship between the length and the period of a pendulum
Repeat Lab 4 with a variety of lengths. Record each length and its corresponding period. (Frequency does not need to be measured here) Verify that the results are consistent with the following equation:
T = 2πL g
Calculate the length required to have a period of 1.0 s. Construct the pendulum to test to your prediction.

25. Why does length affect period?
When two strings have the same angle but different lengths, the arcs that the bob must travel to equilibrium is different and therefore, the periods will be different.
Fig. 12-9

26. Why don’t mass and amplitude affect period?
A heavier mass will provide a large restoring force, but it also needs a larger force to achieve the same acceleration.
Because the acceleration is the same, the periods will be the same.

27. Similarly, the larger amplitude requires a larger restoring force
Similarly, the larger amplitude requires a larger restoring force. The acceleration will be greater but the distance to move is also greater. The period would stay the same.

28. Sample Problem 12B

29. Practice Problems
What is the period of a 3.98 m long pendulum? What is the period of a 99.4 cm long pendulum?

30. Periods of a mass-spring system
Period of a mass-spring system depends on mass and spring constant.
A heavier mass attached to a spring increases inertia without providing a compensating increase in restoring force
A heavy mass has a smaller acceleration than a light mass has. So, a heavy mass has a greater period.
As mass increases, so does the period.

31. The greater the spring constant (k), the stiffer the spring.
A stiffer spring will take less time to complete one cycle of motion than one that is less stiff.
As with the pendulum, changing the amplitude of the vibration does not affect the period.

32. Period of a mass-spring system in simple harmonic motion
T = 2πm k
Period=2π X square root of (mass/spring constant)

33. Sample Problem 12C

34. Practice Problem 12C
A 1.0 kg mass attached to one end of a spring completes one oscillation every 2.0 s. Find the spring constant.

35. Conceptual Challenge
Why is a pendulum a reliable time-keeping device, even if its oscillations gradually decrease in amplitude over time?

36. 12-3: Wave Motion
A wave motion travels away from the disturbance that causes the wave.
Particles in the medium vibrate up and down as the wave passes.
Ex: a leaf in a pond wave
Almost all wave types need a medium to travel through, these are called mechanical waves.
Electromagnetic waves do not need a medium and can travel in outer space
Ex: x-rays, microwaves, etc.

37. Wave Types Pulse waves-a single traveling pulse
Ex: Fig ; a single flip of the wrist while holding one end of a rope that is fixed
Periodic wave-continuous pulses that form periodic motion such as moving your hand up and down repeatedly while holding a rope

38. Fig
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

39. Transverse Waves-particles of the medium vibrate perpendicularly to the motion of the wave
Fig waveform-represents the displacements at a moment in time or the displacements of a single particle as time passes

40. Crest-the highest point above equilibrium
Trough-lowest point below equilibrium
Amplitude-maximum displacement from equilibrium
Fig b
A wave is a cyclical motion, first displaced in one direction, then in the other direction, then returning to equilibrium.

41. Wavelength-the distance the wave travels during one cycle, λ
Crest to crest=wavelength
Trough to trough=wavelength

42. If a spring is pumped back and forth toward the opposite fixed end, a longitudinal wave is formed
Vibrations are parallel to the motion of the wave
The spring should have compressed and stretched regions of the coil that travel along the spring
Ex: sound waves
Fig the compressed regions correspond to the crests and the stretched regions correspond to troughs

43. Compressed regions are high density and high pressure
Stretched regions are low density and pressure

44. Frequency, Period & Wave Speed
Frequency of the vibrations of the source of a wave equal the vibrations of the particles in the wave
Wave frequency describes the number of crests or troughs that pass a given point in a unit of time
Period of a wave describes the time it takes for a complete wavelength to pass a given point
Period is inversely related to frequency

45. A displacement of one wavelength occurs in a time interval equal to one period of the vibration
V=λ/T
Substitute the inverse relationship, f=1/T
Into this equation gives us
V=fλ where the speed of a wave is constant except when traveling from one medium to another

46. Sample Problem 12D

47. Waves carry energy as they move across a medium while the medium doesn’t move
Waves transfer energy by transferring motion of matter rather than by transferring matter itself, they do so efficiently
The greater the amplitude, the more energy carried in a given time
The energy transferred is proportional to the square of the wave’s amplitude

48. Ex: if amplitude of a wave is doubled, the energy is increased by a factor of four
If the amplitude is halved, the energy is decreased by a factor of four

49. 12-4: Wave Interactions
Two mechanical waves can occupy the same space at the same time because they are only displacements of matter, not matter
When two waves pass through one another, they form interference patterns of light and dark bands called superposition, fig
Ex: electromagnetic radiation waves may also interfere

50. Demo 10
Wave superposition

51. Constructive Interference
When two waves are traveling toward each other with the same direction displacements, they form a resultant wave that is equal to the sum of the individual waves, called superposition principle.
If the waves are on the same side of the equilibrium, it is called constructive interference.

52. Once the waves pass through each other, their individual displacements are equal to their previous displacements
Fig

53. Destructive Interference
When the displacements are in opposite directions from the equilibrium, the resultant wave is the difference between the pulses, called destructive interference
Fig

54. Complete Destructive Interference
When two pulses have equal but opposite amplitudes, the resultant wave has a displacement of zero, called complete destructive interference.
Fig
After the interference, each pulse resumes its previous amplitude

55. The superposition principle is true for longitudinal waves as well
A compression is a force on a particle in on direction, while a rarefaction involves a force on the same particle in the opposite direction.

56. Reflection
At a free boundary, waves are reflected due to an upward force at the boundary.
Fig a
At a fixed boundary, waves are reflected and inverted due to a downward force at the boundary.
Fig b

57. Standing Waves
When a string is shaken up and down in a regular motion, it produces waves with the same frequency, wavelength, and amplitude. Those waves are then sent down the string and reflected back down the string toward each other.
The result is a standing wave-a wave pattern that does not move down the string.

58. A standing wave contains both constructive and destructive interference
Fig a
Nodes-points at which two waves cancel, complete destructive interference
Antinodes-point at which the largest amplitude occurs, constructive interference

59. Fig. 12-23 Waves running together create a blur pattern
A single loop represents a crest or trough alone while two loops correspond to a crest and a trough together, one wavelength
The ends of the strings must be nodes, thus it is a standing wave

60. Fig. 12-23 Standing Waves/Wavelength
The wavelengths of these standing waves depends on the string lengths
Wavelengths:
A. String length, L
B. 2L
C. L
D. 2/3L

61. Demo 10-Wave Superposition
Two students hold long coiled spring
One student send a single pulse down the spring
The opposite student then sends an identical pulse.
Both students generate pulses simultaneously.

62. Where do the pulses cross each other?
How can we tell?
How can we tell that the pulses are crossing through each other and not bouncing off each other?
Now, send two pulses of very different amplitudes toward each other.

63. Demo 11-Waves passing each other
Same scenario as demo 10
Create pulses with opposite displacements and observe the waves that reach the hands of the students
Observe the same using waves with different amplitudes and displacements on the same side.
Observe constructive and destructive

64. Demo 12-Wave Reflection
Fix one each of a spring to an object and hold the other in your hand.
Send a pulse down the spring and observe the reflected pulse.
Is the pulse inverted?
Try the same with a piece of rope fixed and then loosely tied to an object.

65. Demo 13-Standing Waves
Create a standing wave with the springs