Vibrations and Waves Chapter 11.

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1 Vibrations and Waves Chapter 11

2 Simple Harmonic Motion
Chapter 11 Section 1

3 Periodic Motion Any repetitive, or cyclical, types of motion
Examples? Simple Harmonic Motion (SHM) is a specialized form of periodic motion

4 Simple Harmonic Motion
Periodic vibration about an equilibrium position Restoring force must be proportional to displacement from equilibrium in the direction of equilibrium

5 Simple Harmonic Motion
Common examples include: mass-spring system pendulum for small angles

6 Mass on a Spring When a spring is stretched, the restoring force from the tension in The spring is described by Hooke’s Law… F = kx The force acting on the mass is proportional to its displacement from equilibrium and in a direction towards equilibrium, thus SHM

7 The Pendulum A simple pendulum consists of a mass called a bob, which is attached to a fixed string. Effectively, all the mass is in the bob. The x component of the weight (Fg sin q) is the restoring force.

8 The Pendulum The magnitude of the restoring force (Fg sin q) is proportional to sin q. When the angle of displacement q is relatively small, sin q is approximately equal to q in radians… sin 0 = 0 So, for small angles, the restoring force is very nearly proportional to the displacement, and the pendulum’s motion is an excellent approximation of simple harmonic motion.

9

10 Virtual Simple Harmonic Motion

11 Measuring Simple Harmonic Motion
Chapter 11 Section 2

12 Amplitude The maximum displacement from equilibrium.

13 Period The time it takes for one complete cycle of motion.
Represented by the symbol T Unit of seconds

14 Frequency The number of cycles completed in a unit of time (usually seconds) Represented by the symbol f Unit of s-1 (also known as Hertz)

15 Period and Frequency f = 1/T and T = 1/f
Period and frequency are inversely related. f = 1/T and T = 1/f

16 A mass-spring system vibrates exactly 10 times each second
A mass-spring system vibrates exactly 10 times each second. What is its period and frequency? f = 10 cycles per second = 10 Hz T = 1/f = 1/10 s = 0.1 s

17 Factors Affecting Pendulums
For small amplitudes, the period of a pendulum does not depend on the mass or amplitude. Length and acceleration due to gravity do affect the period of a pendulum.

18 Factors Affecting Mass-Spring Systems
The heavier the mass, the longer the period (more inertia) The stiffer the spring, the less time it will take to complete one cycle.

19 Chapter 11 Section 3 Properties of Waves

20 What is a wave? A wave is an means by which energy is transferred from one place to another via periodic disturbances

21 Some general terminology…
Pulse – a single disturbance, single cycle Periodic wave – continuous, repeated disturbances Sine wave – a wave whose source vibrates with simple harmonic motion Medium – whatever the wave is traveling through

22 Mechanical Waves Waves that require a physical medium to travel through. Examples: sound, disturbance in a slinky Examples of physical media are water, air, string, slinky.

23 Electromagnetic waves
Waves that do not require a physical medium. Comprised of oscillating electric and magnetic fields Examples include x-rays, visible light, radio waves, etc.

24 Transverse Waves Particles of the medium move perpendicular to the direction of energy transfer You should be able to identify crests, troughs, wavelength (distance traveled during one full cycle), and amplitude Crest Trough

25 Longitudinal Waves Particles of the medium move parallel to the direction of energy transfer (slinky demo) Be able to Identify compressions, rarefactions, wavelengths Compressions Rarefactions

26 Waves transfer energy Note that, while energy is transferred from point A to point B, the particles in the medium do not move from A to B. Individual particles of the medium merely vibrate back and forth in simple harmonic motion The rate of energy transfer is proportional to the square of the amplitude When amplitude is doubled, the energy carried increases by a factor of 4.

27 Wave speed Wave speed is determined completely by the characteristics of the medium For an unchanging medium, wave speed is constant The speed of a wave can be calculated by multiplying wavelength by frequency. v = f x λ

28 Practice #1 Q: Microwaves travel at the speed of light, 3.00108 m/s. When the frequency of microwaves is 9.00 109 Hz, what is their wavelength? A: m

29 Practice #2 Q: The piano string tuned to middle C vibrates with a frequency of 264 Hz. Assuming the speed of sound in air is 343 m/s, find the wavelength of the sound waves produced by the string. A: 1.30 m

30 11.3 Problems Page

31 Wave Interactions Chapter 11 Section 4

32 5 behaviors common to all waves:
Reflection Interference Rectilinear Propagation Refraction Diffraction

33 1. Reflection The bouncing of a wave when it encounters the boundary between two different media

34 Fixed End Reflection At a fixed boundary, waves are inverted as they are reflected.

35 Free End Reflection At a free boundary, waves are reflected on the same side of equilibrium

36 2. Interference The combination of two or more waves in a medium at the same time. Physical matter cannot occupy the same space at the same time, but energy can. The Superposition Principle describes what happens when waves interfere… Waves (energy) pass through each other completely unaffected The medium will be displaced an amount equal to the vector sum of what the waves would have done individually

37 Constructive Interference
Pulses on the same side of equilibrium. Waves meet, combine according to the superposition principle, and pass through unchanged. Displacement of medium greater than originals

38 Destructive Interference
pulses on opposite sides of equilibrium. Waves meet, combine according to the superposition principle, and pass through unchanged. Displacement of medium less than at least one original

39 Complete Destructive Interference

40 Interference patterns
Interference patterns result from continuous interference.

41 Standing Waves An interference pattern that results when two waves of the same frequency, wavelength, and amplitude travel in opposite directions and interfere.

42 Standing wave parts Node – point that maintains zero displacement, complete destructive interference Antinode – point at which largest displacement occurs, constructive interference

43 Standing waves Only specific frequency-wavelength combinations will produce standing wave patterns in a given medium.

44 If a string is 4.0 m long, what are three wavelengths that will produce standing waves on this string?

45 3. Rectilinear Propagation
Waves travel in straight lines The direction of travel is perpendicular to the wavefront. Wavefront - The set of points in space reached by a wave at the same instant as the wave travels through a medium.

46 Parallel Wavefronts: Circular Wavefronts: Direction of a single wave

47 4. Refraction The bending of the path of a wave as it enters a new medium of different wave speed.

48 5. Diffraction The spreading of wave energy around the edges of barriers and obstacles