Vibrations and Waves

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# Vibrations and Waves - PowerPoint PPT Presentation

Vibrations and Waves. Chapter 11. Simple Harmonic Motion. Chapter 11 Section 1. Periodic Motion. Any repetitive, or cyclical, types of motion Examples? Simple Harmonic Motion (SHM) is a specialized form of periodic motion. Simple Harmonic Motion.

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## Vibrations and Waves

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Presentation Transcript

Chapter 11

### Simple Harmonic Motion

Chapter 11 Section 1

Periodic Motion
• Any repetitive, or cyclical, types of motion
• Examples?
• Simple Harmonic Motion(SHM) is a specialized form of periodic motion
Simple Harmonic Motion
• Periodic vibration about an equilibrium position
• Restoring force must be
• proportional to displacement from equilibrium
• in the direction of equilibrium
Simple Harmonic Motion
• Common examples include:
• mass-spring system
• pendulum for small angles
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

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.

The Pendulum
• The magnitude of the restoring force (Fgsin 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 ofsimple harmonic motion.
Virtual Simple Harmonic Motion
Amplitude
• The maximum displacement from equilibrium.
Period
• The time it takes for one complete cycle of motion.
• Represented by the symbol T
• Unit of seconds
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)
Period and Frequency
• Period and frequency are inversely related.
• f = 1/T and T = 1/f
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

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.
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.
Properties of Waves

Chapter 11 Section 3

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

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.
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.
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

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

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.
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

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: 0.0300 m
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
11.3 Problems
• Page 387 1-4
Wave Interactions

Chapter 11 Section 4

5 behaviors common to all waves:
• Reflection
• Interference
• Rectilinear Propagation
• Refraction
• Diffraction
1. Reflection
• The bouncing of a wave when it encounters the boundary between two different media
Fixed End Reflection
• At a fixed boundary, waves are inverted as they are reflected.
Free End Reflection
• At a free boundary, waves are reflected on the same side of equilibrium
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
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
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
Interference patterns
• Interference patterns result from continuous interference.
Standing Waves
• An interference pattern that results when two waves of the same frequency, wavelength, and amplitude travel in opposite directions and interfere.
Standing wave parts
• Node – point that maintains zero displacement, complete destructive interference
• Antinode – point at which largest displacement occurs, constructive interference
Standing waves
• Only specific frequency-wavelength combinations will produce standing wave patterns in a given medium.
If a string is 4.0 m long, what are three wavelengths that will produce standing waves on this string?
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.

Parallel Wavefronts:

Circular Wavefronts:

Direction of a single wave

Direction of a single wave

4. Refraction

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

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