 # Vibrations and Waves Chapter 11.

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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 is a specialized form of periodic motion

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

Restoring Force The push or pull that brings the mass back towards equilibrium The restoring force of a pendulum is a component of the bob’s weight. The restoring force for a mass-spring system is from the stretch (or compression) of the spring

Simple Harmonic Motion
Common examples include a mass-spring system or a pendulum For a pendulum, SHM only for small angles (within 10 degrees of vertical)

Describe speed, acceleration, and restoring force at each point.
Relaxed Length Describe speed, acceleration, and restoring force at each point.

Describe speed, acceleration, and restoring force at each point.

Virtual Simple Harmonic Motion

Measuring Simple Harmonic Motion
Chapter 11 Section 2

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

A mass spring system completes 10 cycles each second.
What is the period? 1/10 s What is the frequency? 10 cycles per second (10 Hz)

Factors Affecting Pendulums
For small amplitudes, the period of a pendulum does not depend on the mass or amplitude. Length does 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.

11.2 Problems Page 379 all Page 381 all except for #3 on Section Review

Chapter 11 Section 3 Properties of Waves

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

Wave Motion A wave is the motion of energy away from a source of periodic disturbance. Mechanical waves require a physical medium to travel through. Examples: sound, disturbance in a slinky Examples of physical media are water, air, string, slinky.

Electromagnetic waves
Do not require a physical medium. 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 Calculate speed of a wave 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: 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 Page

Wave Interactions Chapter 11 Section 4

Interference The combination of two or more waves in a medium at the same time. 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
Waves are on the same side of equilibrium. Waves meet, combine according to the superposition principle, and pass through unchanged. Amplitude larger than originals

Destructive Interference
Waves are on the opposite sides of equilibrium. Waves meet, combine according to the superposition principle, and pass through unchanged. Amplitude smaller than at least one original wave

Complete Destructive Interference

Interference patterns
Interference patterns result from continuous interference. Check it out!

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

Standing Waves A wave 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
Antinode – point at which largest displacement occurs

Standing waves Only certain frequencies produce standing wave patterns.

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