Transcript File

Chapter 11: Vibrations and Waves
Periodic Motion
• When a vibration (aka oscillation)
repeats itself back and forth over the same
path the motion is called periodic.
• One of the most common examples of this is
an object bouncing at the end of a spring, so
we will look at this example closely.
Spring – Mass Systems
• Assume a massless spring attached sideways
with an object of mass m attached to one end.
• All springs will have a relaxed length where
the mass will just sit still, this is called the
equilibrium position.
Choices Choices
• There can be 3 possible scenarios.
– The mass could be at the equilibrium position.
– The mass could be between the equilibrium
position and the other end of the spring (the
spring is squished).
– The mass could be beyond the equilibrium
position (the spring is stretched).
You’re making me uncomfortable
• A spring hates to be away from its equilibrium
position.
• If moved away from its EP, a spring will try to
move back to it.
• Remember chapter 6, springs create a force
equal to spring constant times displacement.
F = -kx
A sad truth
• So let’s say we have a mass on a spring and we
squish it in a little bit then let go.
• The spring puts a force on the mass in the
opposite direction we pushed it, trying to get
back to EP.
• The spring over shoots the EP and exerts a
force pulling the mass back to EP, but it over
shoots it again!
Warning: Vocab
• The distance we move the mass away from EP is
called the displacement.
• The maximum displacement is called the
amplitude.
• One complete back and forth motion is called a
cycle.
• Period is the time per cycle and frequency
is the number of cycles per time.
Simple Harmonic Motion
• Any vibrating system for which the restoring
force is directly proportional to the negative of
the displacement is said to exhibit simple
harmonic motion.
– Simple: force is caused only by displacement, it’s
not rocket propelled.
– Harmonic: continuous back and forth movement.
– Motion: duh
Simple Harmonic Oscillator
• A system that creates simple harmonic motion
is called a simple harmonic oscillator.
• Today we examine the energies involved in
such a system.
• Relax, it’s just a review of chapter 6.
Getting things started
• The simplest simple harmonic oscillator is our
mass and spring again.
• To get the system bouncing we first need to
squish or stretch the spring.
• When we do that we give the spring potential
energy, remember PE = 1/2kx2 (chapter 6)
The return of conservation of energy
• In chapter 6 we learned that the total energy
of a system, E, is always equal to the kinetic
energy plus the potential energy.
• E = KE + PE
• E = 1/2mv2 + 1/2kx2
Extreme Measures
• It is simplest to look at the extreme points of
the system to find its total E.
• At the maximum displacement (called
amplitude, A) all of E is PE
E = 1/2m(0)2 + 1/2kA2 = 1/2kA2
• At the equilibrium position, EP, all of E is KE
E = 1/2mv02 + 1/2k(0)2 = 1/2mv02
• Note: in this chapter v0 is the MAX velocity.
The Period of Simple Harmonic
Motion
• V0 = 2A
T
• Solving for T gives us T =
2A
v0
• Remember 1/2kA2 = 1/2mv2, so A/v0 = √(m/k)
m
• So T = 2
k
Frequency of SHM
• Because f = 1/T,
• f= 1 k
2
m
Position as a function of time
• How can we figure out the distance our object
is from the equilibrium point at any given
time?
• x = A cos ωt
• x = A cos 2πft
• x = A cos (2πt / T)
Velocity and acceleration as a
function of time
• v = -v0 sin(2πt / T)
• a = -(kA/m) cos(2πft) = -a0 cos(2πt / T)
Wave Motion
• In Chapters 11 and 12 we are only concerned
with one family of waves, mechanical waves.
• Mechanical waves are waves created by
mechanical forces
– Shaking a slinky
– An earthquake
– A car going over a bump
A common misconception
• Many people think that waves carry matter
like surfing.
• This is NOT true
• A wave is energy, a chain reaction.
Types of waves
• Transverse wave – a wave that travels
perpendicular to the vibrations
• Longitudinal wave – a wave that travels
parallel to the vibrations
– Aka density wave or pressure wave
Vocab
• Pulse – a single wave bump. (What you made
with the slinkies)
• Continuous/Periodic wave – when the force
making the wave is a vibration. (What you did
when you shook the slinky constantly)
• Amplitude – the max height of the wave
Even more vocab
• Wavelength (λ) the distance from peak to
peak.
• Frequency (f) – the number of peaks per unit
time.
• Wave Velocity (v) – the velocity that wave
peaks move.
The speed of waves
•
•
•
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What is speed?
For a wave, what is distance? Time?
So v = λ / T
Because T = 1/f, we can write
v = λf
speed of a wave =
frequency x wavelength
Speed Limit
• Remember the slinky lab?
• What happened as frequency increased?
• The speed of a mechanical wave for any
given medium is fixed.
Superposition of Waves
• Last time we learned different sound
waves in the same medium travel at the
same speed.
• So two different sounds played at the
same time and distance reaches your ear
at the same time.
• How can 2 waves be in the same place at
the same time?
Superposition of Waves 2
• Waves are not matter, they are
displacements of matter.
• They can occupy the same space.
• The combination of two overlapping
waves is called superposition.
Interference
• When two waves overlap they have an
effect on each other.
• This effect can be observed by studying
the interference pattern of the waves.
• We will first look at the two extreme
cases of interference.
Constructive Interference
• Superposition principle – the amplitude
of the resulting wave can be found by
adding the amplitudes of each wave.
• When the displacement is on the same
side for each, the sign is the same and
the wave is bigger. This is constructive
interference
Destructive Interference
• When the displacement of two
overlapping waves are in opposite
directions their signs are different.
• So, when added they produce a smaller
wave. This is destructive interference.
• When 2 equal and opposite waves
overlap, their sum is zero. This is called
complete destructive interference.
Reflection of waves
• If you shook your slinky hard enough you
saw it start to come back.
• If waves are not matter then how do they
bounce off things?
Newton’s Third strikes again
• Picture a string tied to a pole so that the
knot can move freely
• As the pulse travels down the string the
knot moves up.
• Tension pulls the knot back down
creating a disturbance in the rope.
• This creates a new wave back in the same
direction.
If the end was fixed
• If the string fixed to the pole instead how
would it change things?
• When the pulse hits the pole it wants to
move up, but the pole exerts a
downward force.
• This force creates a wave that is in the
opposite displacement of the original
wave.
Standing waves
• In the last slide, what would happen if
the string was moved up and down not
just once but constantly?
• This
• This is what is known as a standing wave
Nodes and Antinodes
• Nodes (N)- places where complete destructive
interference occurs.
• Antinodes (A) – the midway point between
two nodes. The amplitude is highest at this
point.
More on standing waves
• Only particular frequencies, and
therefore, particular wavelengths can
produce standing waves.
• The wavelength of a standing wave
depends on the length of the string.
Standing Waves
• Yesterday we ended class by observing the
standing wave.
• The points of destructive interference, where
the string stays still, are called nodes.
• The points of constructive interference, where
the string reaches its max height, are called
antinodes.
All Natural
• A standing wave can occur at more than one
frequency.
• However, they can only occur at specific
frequencies.
• The frequencies that do create standing waves
are called the natural frequencies or
resonant frequencies of the string.
Harmonics
• The wavelengths of the natural frequencies
depend on the length of the string itself.
• The lowest frequency, called the
fundamental frequency, has a
wavelength given by the following:
L = ½λ1
Overtones
• The other natural frequencies are called
overtones.
• The wavelengths of each overtone can be
found using the following:
L = nλn / 2
• Or λn = 2L / n
Frequency
• From v = λf we know f = v/ λ
• So the frequency of each harmonic can be
found using:
fn = v / λn = nv / 2L
• Finally, v =√(FT / (m/L))
• With these tools we can know exactly how
long to make certain strings so that they make
specific noises. This is what makes music
possible!