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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 (SHM) is a specialized form of periodic motion. Simple Harmonic Motion.

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simple harmonic motion

Simple Harmonic Motion

Chapter 11 Section 1

periodic motion
Periodic Motion
  • Any repetitive, or cyclical, types of motion
    • Examples?
  • Simple Harmonic Motion(SHM) is a specialized form of periodic motion
simple harmonic motion1
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 motion2
Simple Harmonic Motion
  • Common examples include:
    • mass-spring system
    • pendulum for small angles
mass on a spring
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
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 pendulum1
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
Virtual Simple Harmonic Motion
  • http://phet.colorado.edu/simulations/sims.php?sim=Pendulum_Lab
  • http://phet.colorado.edu/simulations/sims.php?sim=Masses_and_Springs
amplitude
Amplitude
  • The maximum displacement from equilibrium.
period
Period
  • The time it takes for one complete cycle of motion.
  • Represented by the symbol T
  • Unit of seconds
frequency
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
  • 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
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
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
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
Properties of Waves

Chapter 11 Section 3

what is a wave
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
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
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
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
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
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
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
  • 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
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
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
11.3 Problems
  • Page 387 1-4
wave interactions
Wave Interactions

Chapter 11 Section 4

5 behaviors common to all waves
5 behaviors common to all waves:
  • Reflection
  • Interference
  • Rectilinear Propagation
  • Refraction
  • Diffraction
1 reflection
1. Reflection
  • The bouncing of a wave when it encounters the boundary between two different media
fixed end reflection
Fixed End Reflection
  • At a fixed boundary, waves are inverted as they are reflected.
free end reflection
Free End Reflection
  • At a free boundary, waves are reflected on the same side of equilibrium
2 interference
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
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
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
  • Interference patterns result from continuous interference.
  • http://phet.colorado.edu/en/simulation/wave-interference
standing waves
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
Standing wave parts
  • Node – point that maintains zero displacement, complete destructive interference
  • Antinode – point at which largest displacement occurs, constructive interference
standing waves1
Standing waves
  • Only specific frequency-wavelength combinations will produce standing wave patterns in a given medium.
slide44
If a string is 4.0 m long, what are three wavelengths that will produce standing waves on this string?
3 rectilinear propagation
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.

slide46

Parallel Wavefronts:

Circular Wavefronts:

Direction of a single wave

Direction of a single wave

4 refraction
4. Refraction

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

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