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 massspring 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.
Describe
speed,
acceleration,
and restoring
force at each
point.
Virtual Simple Harmonic Motion
http://phet.colorado.edu/simulations/sims.p
hp?sim=Pendulum_Lab
http://phet.colorado.edu/simulations/sims.p
hp?sim=Masses_and_Springs
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
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
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
Calculate speed of a wave by multiplying
wavelength by frequency.
–v=fxλ
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
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
pulses on 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 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
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?