1. Simple Harmonic Motion Simple harmonic motion (SHM) refers to a certain kind of oscillatory, or wave-like motion that describes the behavior of many physical phenomena: –a pendulum –a bob attached to a spring –low amplitude waves in air (sound), water, the ground –the electromagnetic field of laser light –vibration of a plucked guitar string –the electric current of most AC power supplies

2. SHM Position, Velocity, and Acceleration What do these slopes tell you?

3. Simple Harmonic Motion Periodic Motion: any motion of system which repeats itself at regular, equal intervals of time.

4. Simple Harmonic Motion

5. Equilibrium: the position at which no net force acts on the particle. Displacement: The distance of the particle from its equilibrium position. Usually denoted as x(t) with x=0 as the equilibrium position. Amplitude: the maximum value of the displacement with out regard to sign. Denoted as x max or A.

6. The period and frequency of a wave the period T of a wave is the amount of time it takes to go through 1 cycle the frequency f is the number of cycles per second –the unit of a cycle-per-second is commonly referred to as a hertz (Hz), after Heinrich Hertz (1847-1894), who discovered radio waves. frequency and period are related as follows:

7. Restoring Force Simple harmonic motion is the motion executed by a particle of mass m subject to a force that is proportional to the displacement of the particle but opposite in sign. How does the restoring force act with respect to the displacement from the equilibrium position? F is proportional to –x (F=kx)

8. Springs and Simple Harmonic Motion

9. Equations of Motion Conservation of Energy allows a calculation of the velocity of the object at any position in its motion…

10. Conservation of Energy For A Spring in Horizontal Motion E = Kinetic + Elastic Potential E = ½ mv 2 + ½ kx 2 = Constant At maximum displacement, velocity is zero and all energy is elastic potential, so total energy is equal to ½ kA 2

11. Energy to velocity

12. Proofs to determine time

13. Period (T) for springs The period of the spring is the time it takes for the motion of the mass going through 1 complete cycle It is affected by mass and K (restoring forces)

16. 2. The period of a spring-mass system undergoing simple harmonic motion is T. If the amplitude of the spring-mass system’s motion is doubled, the period will be: (A) ¼ T (B) ½ T (C) T (D) 2T (E) 4T

19. 8. Which of the following is true for a system consisting of a mass oscillating on the end of an ideal spring? (A) The kinetic and potential energies are equal to each other at all times. (B) The kinetic and potential energies are both constant. (C) The maximum potential energy is achieved when the mass passes through its equilibrium position. (D) The maximum kinetic energy and maximum potential energy are equal, but occur at different times. (E) The maximum kinetic energy occurs at maximum displacement of the mass from its equilibrium position

20. 19. A block attached to the lower end of a vertical spring oscillates up and down. If the spring obeys Hooke's law, the period of oscillation depends on which of the following? I. Mass of the block II. Amplitude of the oscillation III. Force constant of the spring (A) I only (B) II only (C) III only (D) I and II (E) I and III

21. 28. Two objects of equal mass hang from independent springs of unequal spring constant and oscillate up and down. The spring of greater spring constant must have the (A) smaller amplitude of oscillation (B) larger amplitude of oscillation (C) shorter period of oscillation (D) longer period of oscillation (E) lower frequency of oscillation

22. 38. A 1.0 kg mass is attached to the end of a vertical ideal spring with a force constant of 400 N/m. The mass is set in simple harmonic motion with an amplitude of 10 cm. The speed of the 1.0 kg mass at the equilibrium position is (A) 2 m/s (B) 4 m/s (C) 20 m/s (D) 40 m/s (E) 200 m/s

23. Pendulums When we were discussing the energy in a simple harmonic system, we talked about the ‘springiness’ of the system as storing the potential energy But when we talk about a regular pendulum there is nothing ‘springy’ – so where is the potential energy stored?

24. The Simple Pendulum As we have already seen, the potential energy in a simple pendulum is stored in raising the bob up against the gravitational force The pendulum bob is clearly oscillating as it moves back and forth – but is it exhibiting SHM?

25. F=FgsinO is the restoring force. Since F is proportional to sinO and not O itself. (meaning not SHM) But if the angle is small enough than the Percentage difference lower than 2.0 % Difference.

26. The Simple Pendulum The period of a pendulum is given by: If all of the mass of the pendulum is concentrated in the bob and we get:

27. We will prove this.

28. 6. What is the period of a simple pendulum if the cord length is 67 cm and the pendulum bob has a mass of 2.4 kg. (A) 0.259 s (B) 1.63 s (C) 3.86 s (D) 16.3 s (E) 24.3 s

29. 7.If the mass of a simple pendulum is doubled but its length remains constant, its period is multiplied by a factor of (A).5 (B).707 (C) 1 (D) 1.41 (E) 2

30. 9. The length of a simple pendulum with a period on Earth of one second is most nearly (A) 0.12 m (B) 0.25 m (C) 0.50 m (D) 1.0 m (E) 10.0 m

32. 21. A simple pendulum of length l, whose bob has mass m, oscillates with a period T. If the bob is replaced by one of mass 4m, the period of oscillation is (A).25 T (B).5 T (C) T (D) 2T (E)4T

33. 32. A pendulum with a period of 1 s on Earth, where the acceleration due to gravity is g, is taken to another planet, where its period is 2 s. The accel­eration due to gravity on the other planet is most nearly (A) g/4 (B) g/2 (C) g (D) 2g (E) 4g

34. Looks as waves

35. Day 2

36. Resonance Resonance occurs when small forces are applied at regular intervals to a vibrating or oscillating object and the amplitude of the vibration increases. Kid on a swing How could this be applied to springs/pendulums? Could this create repeating waves?

37. Wave Properties A wave is a disturbance that carries energy through matter or space. Water waves, sound waves, and the waves that travel down a rope or spring are types of mechanical waves.

38. Transverse Waves  A wave pulse is a single bump or disturbance that travels through a medium.  If the wave moves up and down at the same rate, a periodic wave is generated.  A transverse wave is one that vibrates perpendicular to the direction of the wave’s motion.

39. Longitudinal Waves A longitudinal wave the disturbance is in the same direction as, or parallel to, the direction of the wave’s motion. Sound waves are longitudinal waves. Fluids usually transmit only longitudinal waves.

41. Measuring a Wave Speed How fast does a wave move? –Measure the displacement of the wave peak,  d, then divide this by the time interval,  t, to find the speed, given by v =  d/  t. For most mechanical waves, both transverse and longitudinal, the speed depends only on the medium through which the waves move.

43. Amplitude The amplitude of a wave is the maximum displacement of the wave from its position of rest or equilibrium. Amplitude depends on how the wave is generated, but not on its speed.

44. Wavelength Each low point, called a trough, and each high point, called a crest, of a wave. The shortest distance between points where the wave pattern repeats itself is called the wavelength. The Greek letter lambda,,represents wavelength.

45. Amplitude & Wavelength

46. Phase Any two points on a wave that are one or more whole wavelengths apart are in phase. –Particles in the medium are said to be in phase with one another when they have the same displacement from equilibrium and the same velocity. –Particles in the medium with opposite displacements and velocities are 180° out of phase. A crest and a trough, for example, are 180°out of phase with each other. Two particles in a wave can be anywhere from 0° to 180° out of phase with one another.

47. Period and Frequency Period, , and frequency, f, apply only to periodic waves. They do not depend on the wave’s speed or the medium. The frequency of a wave, f, is the number of complete oscillations it makes each second it is measured in hertz. One hertz (Hz) is one oscillation per second (1/s). Frequency of a Wave f = 1/  The frequency of a wave is equal to the reciprocal of the period. Wavelength = v/f The wavelength of a wave is equal to the velocity divided by the frequency.

48. 4. A wave has a frequency of 50 Hz. The period of the wave is: A) 0.010 s B) 0.20 s C) 7 s D) 20 s E) 0.020 s

49. 5. If the frequency of sound is doubled, the wavelength: A) halves and the speed remains unchanged B) doubles and the speed remains unchanged C) is unchanged and the speed doubles D) is unchanged and the speed halves E) halves and the speed halves

50. Speed of sound in air 10. An observer hears a sound with frequency 400 Hz. Its wavelength is approximately A) 0.85 m C) 1.2 m C) 2.75 m D) 13.6 m E) 44 m

51. Velocity for waves on Strings Velocity for waves on

52. Now the ideas For Strings, faster V for waves if: More tension Density of string is lower Can you explain?

53. Example 11-11 A wave whose wavelength is.30 m is traveling down a 300 m long string whose mass is 15 kg. If the wire is under 1000 N of tension, what is the speed and frequency of the wave? 140 m/s and 470 Hz

54. Wave Behavior Boundaries –Reflected –Pass through the boundary –Change direction Reminder: –Velocity depends on the medium Water wave depth of the water affects wave speed Sound waves in air, temperature affects wave speed Waves in a spring, depends on spring tension and mass per unit length

57. Boundaries and Waves Incident Wave: wave that strikes the boundary Reflected Wave: wave that returns in the larger spring For a wall boundary, if little energy is transmitted into the wall the wave may have the same amplitude, but be inverted

58. Superposition of Waves The principle of superposition states that the displacement of a medium caused by two or more waves is the algebraic sum of the displacements caused by the individual waves. Two are more waves can combine to form a new wave.

59. Wave Interference The result of the superposition of two or more waves is called interference. –Constructive interference occurs when the waves are in sync (in phase), and they combine –Destructive interference occurs when the waves are out of sync, and the cancel some or all of each other out.

60. Constructive Interference

61. Destructive Interference

66. Not all of chap 11 Part of 11 will be combined with 12