Transcript wave

Chapter 11
Vibrations & Waves
11.1 Simple Harmonic Motion
Hooke’s Law
Repeated motion is “periodic motion”.
Like a pendulum, back and forth over
the same path.
At equilibrium position, speed reaches maximum
As the mass is pulled away, it displaces the spring to a
certain distance, x = ?
This exerts a force towards the equilibrium position.
At (b) force = 0 and so does displacement.
Acceleration also equals “0”.
However, speed is at max due to momentum.
At (c) due to momentum, mass overshoots equilibrium
and compresses spring the other way.
At maximum displacement, spring force and acceleration
reach a maximum
Beyond equilibrium the force and acceleration increase,
however speed slows down.
At maximum displacement, speed is 0 but acceleration
and force are at max. Then, back and forth (oscillates).
Friction brings the vibrating mass to rest (damping).
In simple harmonic motion, restoring force
is proportional to displacement
This pushing and pulling is sometimes called
“restoring force”, it is directly proportional to the
displacement of a mass.
Determined by Robert Hooke in 1678…
Felastic  kx
The negative sign means that the spring force is
opposite the direction the mass is displaced.
Harmonic Motion Animation
k is the spring constant and is based on the stiffness of
the spring in N/m.
The motion of the vibrating mass is an example of
“simple harmonic motion” and any periodic motion that is
a result of a restoring force.
Spring Constant Animation
Practice A
Hooke’s Law
A mass of 0.55 kg, attached to a vertical
spring, stretches the spring -0.020 m from its
original equilibrium position.
What is the spring constant?
Given:
m = 0.55 kg
g = 9.81 m/s2
x = 0.020 m
Unknown: k = ?
Answer
270 N/m
A stretched or compressed spring has
elastic potential energy
A bent bow is the same as a stretched
spring.
Stretched or compressed springs store elastic potential
energy (PE).
Once released the PE becomes KE, or moving arrow!
Inside
The Simple Pendulum
The swinging motion of a pendulum is periodic
vibration.
A simple pendulum consists of a mass called
a bob which is attached to a fixed string.
The restoring force of a pendulum is a component
of the bob’s weight
If the restoring force is proportional to the displacement,
then the pendulum’s motion is simple harmonic.
Any displacement from equilibrium can be
resolved with both the x and y components.
Simple Pendulum Animation
For small angles, the pendulum’s motion is simple harmonic
When the maximum angle for displacement θ is relatively
small (<150), sin θ is approximately equal in radians.
Pendulum's motion is an excellent approximation of simple
harmonic motion.
Because a simple pendulum
vibrates with simple harmonic
motion, many of our earlier
conclusions for a mass-spring
system apply here.
Gravitational potential increases as a pendulum’s
displacement increases
This diagram shows how a pendulum’s mechanical
energy changes as the pendulum oscillates.
As the pendulum swings toward equilibrium, it gains
kinetic energy and losses potential energy.
Questions
periodicmotion.
1. Repeated motion is _______
2. When the mass/spring is at maximum displacement,
speed is 0 but acceleration and force are at _____.
max
3. The motion of a vibrating mass is an example of “simple
harmonic motion” and any periodic motion that is a result of a
restoring force.
_________
gravity
4. The restoring force of a pendulum’s motion is _______
5. As the pendulum swings toward equilibrium, it gains
potentialenergy.
kinetic energy and losses ________
11.2
Measuring Simple Harmonic Motion
Amplitude, Period, and
Frequency
A moving trapeze always returns to the
same displacement from equilibrium,
this is the amplitude.
A pendulum’s amplitude can be
measured by the angle between the
pendulum’s equilibrium position and its
maximum displacement.
For a mass-spring system, this is its
stretched or compressed position.
Period and frequency measure time
From one side of max displacement to the
other, one complete cycle, is period, T
If one complete cycle takes 20 seconds, then
the period of motion is 20 s.
The number of complete cycles in a unit of time
is frequency.
If it takes 20 s to complete one cycle, the
frequency is 1/20 cycles or 0.05 cycles.
Frequency is s-1 or hertz (Hz)
So…
-period is time per cycle
-frequency is number of cycles
per unit of time.
1
T
f
1
f 
T
Animation
The period of a simple pendulum depends on
pendulum length and free-fall acceleration
Simple pendulums and mass-spring systems vibrate with
harmonic motion.
To calculate period (T) and frequency ƒ in (Hz), requires
a separate formula.
A pendulum with the same length (L) but with bobs of
two different masses or amplitude, the period will be the
same.
If free-fall acceleration (gravity) changes,
so will the period.
L
T  2
g
When two pendulums have different lengths but the
same amplitude, the shorter pendulum will have a
smaller arc to travel through.
Mass and amplitude do not affect the period for the
same reason all objects fall at the same rate.
The reason for this is, the more you increase the
amplitude, the more restoring force there is (even though
it has a greater distance to cover)
Practice B Simple Harmonic Motion of
a Simple Pendulum
You need to know the height of a tower, but darkness
obscures the ceiling.
You note that a pendulum extending from the ceiling
almost touches the floor and that it has a period of 12 s.
How tall is the tower?


Answer 36 m
Period of a mass-spring system depends on mass
and spring constant
The period of a mass-spring system uses Hook’s Law.
Felastic  kx
Because heavier objects have more inertia, they take
longer to speed up.
This causes them to have
longer periods.
The greater the spring constant (k), the greater the force
needed to stretch or compress the spring.
When force is great, so is the acceleration.
This makes the time required to make one cycle less.
So, stiff spring = short period.
m
T  2
k
Also, changing amplitude does not effect the period.
Practice C Simple Harmonic Motion of
a Mass-Spring System
The body of a 1275 kg car is supported
on a frame by four springs. Two people
riding in the car have a combined mass of 153 kg.
When driven over a pothole in the road, the frame
vibrates with a period of 0.840 s.
For the first few seconds, the vibration
approximates simple harmonic motion.
Find the spring constant of a single spring.
Answer
20,000 N/m
Questions
amplitude can be measured by the angle
1. A pendulum’s _________
between the pendulum’s equilibrium position and its
maximum displacement.
2. The number of complete cycles in a unit of time is known
frequency
as _________.
3. A pendulum with the same length but with bobs of two
period will be the same.
different masses or amplitude, the ______
4. Mass and amplitude (do / do not) affect the period of a
pendulum.
do not
5. The stiffer the spring, the (longer / shorter) the period.
shorter
11.3 Properties of Waves
Wave Motion
If there is a disturbance in a pond, you see a circular
pattern move outward in all directions.
If there is a leaf floating near by, it moves a little but does
not travel with the wave.
Breaking waves are different 
A wave is the motion of a disturbance
This disturbance causes the pattern to
move out in a circular pattern.
Water in the pond is the medium
through which the disturbance travels.
The medium (water) does not actually
travel with the wave.
Sound waves require air as their
medium. In space there is no sound.
Waves that require a material medium
are called mechanical waves.
Animation
Wave Types
A wave that consists of a single traveling pulse is called
a pulse wave.
If you continue to generate pulses, this will create a
periodic wave.
Sine waves describe particles vibrating with simple
harmonic motion
Periodic waves can show simple harmonic motion on a
string.
A wave whose source vibrates with simple harmonic
motion is called a sine wave.
These are called sine waves because a graph of
trigonometric function y = sin x produces this curve when
plotted.
Vibrations of a transverse wave are perpendicular
to the wave motion
When vibrations are perpendicular to the direction of the
wave’s motion they are called transverse waves.
Displacement of a single particle as time passes creates
a wave form.
Wave measures include crest, trough, amplitude,
and wavelength
A wave can be measured in terms of its displacement
from equilibrium.
The highest point is called the crest.
The lowest point the trough.
Remember, amplitude is a measure of maximum
displacement from equilibrium.
The distance the wave travels in one cycle along its path
is called wavelength, (λ).
Vibrations of a longitudinal wave are parallel to the
wave motion
When the displacement of the medium vibrates parallel
to the direction of wave motion is called a longitudinal
wave.
Animation
Longitudinal waves can also be described by a sine
curve.
The type of wave represented above is often called a
density or pressure wave.
The crests are where the spring coils are compressed.
Period, Frequency, and Wave Speed
Sound waves may begin with the
vibrations of your vocal cords. The
source of wave motion is a
vibrating object.
When the vibrating particles of the
medium complete one full cycle,
one wavelength passes any given
point.
Thus, wave frequency describes
the number of waves that pass a
given point in a unit of time.
The period of a wave is the time
required for one complete cycle of
vibration of the mediums particles.
Wave speed equals frequency times wavelength
We can now derive an expression for the speed of a
wave in terms of its period or frequency.
Animation
The speed of a mechanical wave is
constant for any given medium.
Even though all sounds are different,
they reach your ears at the same speed.
As a result, when the frequency increases
its wavelength must decrease.
Practice D
Wave Speed
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.

Answer
1.30 m
Waves transfer energy
Waves transfer energy by the vibration of matter rather
than by the transfer of matter itself.
For this reason, waves are often able to transport energy
efficiently.
The rate at which a wave transfers energy depends on
the amplitude, the greater the amplitude, the more
energy carried in a given time interval.
When the amplitude is doubled, the
energy carried increases by a factor of 4.
Questions
1. Waves that require a material medium are called
mechanical waves.
__________
2. A wave whose source vibrates with simple harmonic
sine wave.
motion is called a _____
3. When the displacement of the medium vibrates parallel
longitudinal
to the direction of wave motion is called a __________
wave.
4. When the frequency increases, its wavelength must?
decrease.
5. When the amplitude is doubled, the
four
energy carried increases by a factor of ______.
11.4
Wave Interactions
Wave Interference
When two waves come together, they do not bounce
back like bumper boats.
With sound waves, you can distinguish the sounds of
different instruments.
This is because sound waves (mechanical waves) are
not matter but displacements of matter.
Two waves can occupy the same space
at the same time.
As they pass through one another, they
interact to form an interference pattern.
Displacements in the same direction produce
constructive interference
When two pulses meet, a resultant wave forms.
The amplitude of the resultant wave is equal to the sum of
the amplitudes of each pulse.
Summing the displacements of waves is known as the
superposition principle.
After the two pulses pass, they have their original shape.
If the displacements are on the same side of equilibrium,
when added together, we get constructive interference.
Displacements in opposite directions produce
destructive interference
The following shows what happens when pulses are on
opposite sides of equilibrium.
When the positive and negative displacements are
added we get destructive interference.
When two pulses coincide, their resultant wave can have
a displacement of zero.
This is known as complete destructive interference.
Animation
The superposition principle is valid for longitudinal
(compression) waves.
In a compression, particles move closer together, while
rarefaction, particles spread apart.
When a compression and rarefaction interfere, there is
destructive interference.
In the case of sound, these wave can
cancel causing a lack of sound.
Reflection
At a free boundary, waves are reflected
At a free boundary, the rope is
free to move up and down sending
the wave back in the same way.
This is called reflection.
At a fixed boundary, waves are reflected and inverted
When the pulse reaches the
wall, the pulse exerts an
upward force on the wall.
The wall in turn exerts an
equal and opposite reaction
force on the rope.
As a result, the pulse is
inverted.
Standing Waves
Standing waves occur when a string is
attached to one ridged end and shaken
in a regular motion.
This will produce a wave of a certain
frequency, wavelength, and amplitude
traveling down the string.
As the waves reach the other end they
are reflected back toward the oncoming
waves.
If the string is vibrated at a certain
frequency, a standing wave or resultant
wave pattern appears on the string.
The standing wave consists of alternating
regions of constructive and destructive
interference.
Standing waves have nodes and antinodes
Four possible standing waves are shown below.
Points where complete destructive interference happen
are called nodes.
Midway between two adjacent nodes, where the string
vibrates with the largest amplitude are the antinodes.
In the second example
below, on the right, shows
where there are 3 nodes(N)
and 2 antinodes(A).
Animation
Only certain frequencies, and
wavelengths, produce standing waves.
A standing wave can only be produced
for any wavelength that allows both
ends of the string to be nodes.
Example (b) is half a wave length, so to
find wave length you multiply the string
length by two (2L).
Example (c) is one wave length so we
just use (L).
Example (d) has 4 nodes and 3
antinodes. We would have to go another
half wavelength to have two full wave
lengths. In this case, to find the length of
one wave, we just multiply by (2/3L).
Questions
1. Sound waves (mechanical waves) are not matter but
displacements of matter.
_____________
2. In a compression, particles move closer together, while
__________,
rarefaction particles spread apart.
3. At a free boundary, the rope is free to move up and
down sending the wave back in the same way, this is called
reflection
_________.
4. Midway between two adjacent nodes, where the string
antinodes
vibrates with the largest amplitude are the _________.
5. If a standing wave is half a wave length, to find its wave
length you multiply the string length by ____.
two
End