1. Simple Harmonic Motion
Vibration around an equilibrium position in which a restoring force is proportional to the displacement from equilibrium.
Back and forth motion over the same path.

2. Simple Harmonic Motion (cont.)
Equilibrium position – starting or resting position (x=0)
Restoring force – the force that returns the mass to the original starting position.

3. Hooke’s Law
Felastic = – k x F: restoring force x: displacement k: spring constant

4. Spring Constant
The spring constant depends on the stiffness of the spring.
Increase stiffness, increase k
negative k indicates that the direction of the restoring force is opposite the mass displacement.

5. Summary of SHM At maximum displacement: the velocity becomes zero.
the acceleration reaches maximum.
the restoring force reaches maximum.
PE is max, KE is zero.

6. Summary of SHM At equilibrium:
the velocity reaches maximum. KE is max.
the acceleration becomes zero.
the restoring force becomes zero.
KE is max, PE is zero.

7. Potential Energy in the System
Mass-spring system
elastic PE.
Simple pendulum
gravitational PE.

8. Measuring Simple Harmonic Motion
Amplitude – maximum displacement from equilibrium.
Pendulum: measured by the angle between equilibrium and displacement.
spring-mass system: the amount the spring is stretched or compressed.

9. Measuring SHM (cont.) Period Frequency
the time it takes to execute one complete cycle of motion.
Frequency
the number of cycles or vibrations per unit of time.
SI Unit: hertz (Hz) f = (or) T = T f

10. The Period of a Simple Pendulum
T = 2   L/g L – length of string g – gravitational pull
The period is independent of the mass and the amplitude.

11. The Period of a Mass-Spring System
T = 2  m/k
A heavy mass will have less acceleration than a light mass increasing the time it takes to complete one cycle.

12. Waves
The motion of a disturbance that propagates through a medium or space.
The transfer of energy without the transfer of matter.

13. Characteristics of Waves
Wavelength – the distance between 2 adjacent similar points of the wave, such as from crest to crest or from trough to trough. ()
Amplitude - maximum displacement on either side of the equilibrium position.

14. Characteristics of Waves (cont.)
Frequency – number of crests or troughs that pass a given point in a unit of time.
Period – amount of time required for one complete wavelength to pass a given point.
Wave speed v = f 

15. Mechanical Waves
A wave whose propagation requires the existence of a medium.
waves result from molecular movement.
energy is needed to start the disturbance.
the medium does not travel with the wave.
After the wave has passed, the particles of the medium return to original position.
Ex: sound (uses air as the medium)

16. Electromagnetic Wave
A wave whose propagation does not need a medium to travel through. (energy waves)
Ex: light waves

17. Waves (cont.) Pulse wave – a single nonperiodic disturbance.
Periodic wave – a wave whose source is some form of periodic motion.
Sine wave – a wave that vibrates with simple harmonic motion. (named for the graph produced when graphing y = sin x).

18. 2Types of Mechanical Waves: Transverse Waves
A wave whose particle displacement is perpendicular to the direction of the wave motion.
Ex: water waves

19. Transverse Waves (cont.)
crest – the highest point above the equilibrium position.
trough – the lowest point below the equilibrium position.
equilibrium line – the resting point.

20. Longitudinal Waves
A wave whose particles vibrate parallel to the direction of the wave motion.
Ex: sound waves
compression – the portion of the wave where particles are pushed together; particles are densely packed.
rarefaction – the portion of the wave behind the compression where the particles are stretched apart;low density of particles.

21. Superposition Principle
Since waves are not matter, but displacement of matter, 2 waves can occupy the same space at the same time.
When 2 or more waves travel through a medium, the resultant wave is the sum of the displacements of the individual waves at each point.

22. Interference
The effects produced by 2 or more waves that superpose while passing through a given region.
2 Types of interference:
Constructive interference
Destructive interference

23. Constructive Interference
Interference in which individual displacements on the same side of the equilibrium position are added together to form the resultant wave.
These waves are said to be “in phase”.
When crest meets crest it increases the amplitude.

24. Destructive Interference
Interference in which individual displacements on opposite sides of the equilibrium position are added together to form the resultant wave.
These waves are said to be “out of phase”
When crest meets trough, the amplitude is canceled.

25. Longitudinal Waves Constructive interference Destructive interference
two compressions interfere to increase the net force on the particles.
Destructive interference
a compression and a rarefaction interfere reducing the net force on the particles.

26. Reflection
When a wave impulse reaches a boundary, the pulse travels back along the medium in the opposite direction.
Incident wave: incoming wave
Reflected wave: returning wave bounced off of a boundary

27. Reflected Wave Pulses Reflected pulse from a free boundary:
The reflected pulse is identical to the incident pulse.
Reflected pulse from a fixed boundary:
The reflected pulse is inverted from the incident pulse.

28. Standing Waves
A wave pattern that results when 2 waves of the same frequency, wavelength, and amplitude travel in opposite directions and interfere.
These 2 waves cancel one another out at the site of destructive interference.
The crest of one wave becomes the trough of the second wave.
The result is the wave doesn’t appear to be moving.

29. Standing Waves (cont.)
Node – point in a standing wave that always undergoes complete destructive interference and therefore is stationary.
No matter disturbance at this point.
Antinode – point in a standing wave, halfway between 2 nodes, at which the largest amplitude occurs. (loops)

30. Doppler Effect
A frequency shift that is the result of relative motion between the source of waves and an observer.
When an ambulance moves toward you with its siren going:
the pitch is higher as it approaches.
the pitch is lower as it moves away.

31. Doppler Effect (cont.)
The pitch depends on frequency, but the frequency from the source of sound is not changing.
The perceived frequency is greater as vehicle approaches
The perceived frequency is lower as vehicle moves away.

32. Bow Wave
The V-shaped wave made by an object moving across a liquid surface at a speed greater than the wave speed.

33. Shock Wave
The cone-shaped wave made by an object moving at supersonic speed through a fluid.
Sonic boom - the loud sound resulting from the incidence of a shock wave.

34. Simple Harmonic Motion Sample Problem 1
If a spring is stretched 2.0 cm by a mass of 0.55 kg, calculate the force constant.

35. Sample Problem 2
A 0.35 kg mass attached to a spring of spring constant 130 N/m is free to move on a frictionless horizontal surface. If the mass is released from rest at x = 0.10m, find the force on it.
Find the acceleration at x = 0.10 m

36. Sample Problem 3
A spring of spring constant 30 N/m is attached to different masses and the system is set in motion. Find the period and frequency of vibration for masses of the following magnitudes: kg g

37. Sample Problem 4
A man needs to know the height of a tower, but darkness obscures the ceiling. He knows, however, that a long pendulum extends from the ceiling almost to the floor and that its period is 12.0s. How tall is the tower?

38. Sample Problem 5
A wave traveling in the positive x direction has a wavelength of 40 cm and an amplitude of 15 cm. Find the period and speed of the wave if it has a frequency of 8.0 Hz.

39. Sample Problem 6
The note middle C on a piano has a frequency of approximately 262 Hz and a wavelength in air of 1.31 m. Find the speed of sound in air.

40. Sample Problem 7
An FM station broadcasts at a frequency of 100 MHz (106), with a radio wave having a wavelength of m. Find the speed of the radio wave.