Transcript Physics Ch. 12 Vibrations and Waves

Physics Ch. 12 Vibrations and
Waves

Any repeated motion over the same path is
called a periodic motion
 Ex: a child swinging on a swing, a
pendulum on a grandfather clock, an
acrobat swinging on a trapeze
 Fig. 12-1 mass attached to a spring
 The spring exerts a force on the mass when
the spring is stretch or compressed.





At unstretched position, the spring is at
equilibrium.
The force decreases as the mass approaches
equilibrium and becomes zero at equilibrium
The mass’s acceleration becomes zero at
equilibrium
The velocity reaches maximum at equilibrium.
The mass’s momentum causes the mass to
overshoot the equilibrium and compress the spring
12-1c: When the spring’s compression is
equal to the stretched distance, the mass is
at maximum displacement and the spring
force and acceleration reach their maximum
 Velocity at this point reaches zero.
 The force works in opposite direction of the
mass’s direction.


In an ideal situation, the mass would vibrate
back and forth indefinitely
 In the real world, friction reduces the
motion and it eventually stops.
 This effect is called damping.

Restoring force-the spring force pushing the
mass back toward its original equilibrium
position.
 Restoring force directly proportional to the
displacement of the mass. (Simple
Harmonic Motion)
Hooke’s Law

Felastic = -Kx

Spring force = -(spring constant X
displacement)

K = spring constant (measure of the
stiffness of the spring)
Sample Problem 12A

If a mass of 0.55 kg attached to a vertical
spring stretches the spring 2.0 cm from its
original equilibrium position, as shown in
fig. 12-2, what is the spring constant?
Practice Problem

A 76 N crate is attached to a spring (k=450
N/m). How much displacement is caused
by the weight of this crate?
Practice Problem

A spring of k=1962 N/m loses its elasticity
if stretched more than 50.0 cm. What is the
mass of the heaviest object the spring can
support without being damaged?
Elastic Potential Energy


A stretched or compressed spring has stored
elastic potential energy.
Ex: an archer that pulls the bowstring back has
only elastic potential energy
 When the bowstring is released, the elastic
potential energy is converted to kinetic energy.
 Because energy is conserved, the elastic potential
energy is converted to kinetic energy of the arrow,
bow, and bowstring.
The Simple Pendulum

A simple pendulum has a mass, called a
bob, attached to a fixed string.
 We assume that the mass of the bob is
concentrated at a point and the mass of the
string is negligible. We also disregard the
effects of friction or air resistance.
 For a physical pendulum, such as an
acrobat, we will assume the same.
Restoring Force

Which force on a pendulum acts as the restoring
force?
 The forces on the pendulum are the force of the
string and the force of the bob’s weight. The force
of the weight can be split into two components, x
and y. The y component is opposite the force of
the string and cancels. The net force on the bob is
the x component of its weight.
 This force then pushes or pulls the motion toward
equilibrium, the restoring force.

The restoring force varies with the position
to the equilibrium
 Decreases as it approaches equilibrium and
becomes zero at equilibrium
 At small angles (<15°) the motion is simple
harmonic

At maximum displacement, the restoring
force and the acceleration are at maximum
while the velocity is zero.
 At equilibrium, the restoring force and
acceleration become zero and velocity
reaches a maximum
 Table 12-1
Conservation of Energy

Spring-elastic potential energy
 Pendulum-gravitational potential energy
 At equilibrium, the gravitational potential
energy is at zero, while the energy is solely
kinetic energy
 At maximum displacement, the kinetic
energy turns entirely to gravitational
potential energy
QuickLab pg. 444

Energy of a Pendulum
12-2: Measuring Simple
Harmonic Motion

Amplitude-maximum displacement from
equilibrium (Can be measured by the angle or by
the amount the spring is stretched or compressed)
 Period (T)-the time it takes a complete cycle of
motion; the pendulum ends of where it started
 Frequency (f)-the number of cycles per second or
the SI unit is Hertz (Hz)

These two units are inversely related:

F=1
T
or T = 1
f
Very Quick Lab-4 Period and
Frequency

Attach the pendulum bob to the string and
suspend the string from the ring stand. Set
the pendulum in motion. Have a student
record the time required to complete 20
oscillations. Then, have another student
record how many times the pendulum bob
returns to the same place each second.

Have students use the first measurement to
find the pendulum’s period. Use T=number
of seconds/20 Ask the students what the
second measurement indicates. Compare
the values.
Period of a pendulum depends on
Length of pendulum

If you have two pendulums with two
different small amplitudes, the periods
would still be the same. Thus, the period
does not depend on the amplitude (<15º).
 However, if the length of the pendulum
string were changed or the free-fall
acceleration, then the period would change.
Period of a simple pendulum
in simple harmonic motion

T = 2πL
g
Period=2π X square root of (length/free-fall
acceleration)
Did you know?

Galileo is credited as the first person to notice that
the motion of a pendulum depends on its length
and is independent of its amplitude (for small
angles). He supposedly observed this while
attending church services at a cathedral in Pisa.
The pendulum he studied was a swinging
chandelier that was set in motion when someone
bumped it while lighting the candles. Galileo is
said to have measured its frequency, and hence its
period, by timing the swings with his pulse.
Very Quick Lab-5-Relationship between
the length and the period of a pendulum

Repeat Lab 4 with a variety of lengths. Record
each length and its corresponding period.
(Frequency does not need to be measured here)
Verify that the results are consistent with the
following equation:
 T = 2πL
g
Calculate the length required to have a period of 1.0
s. Construct the pendulum to test to your
prediction.
Why does length affect
period?

When two strings have the same angle but
different lengths, the arcs that the bob must
travel to equilibrium is different and
therefore, the periods will be different.

Fig. 12-9
Why don’t mass and
amplitude affect period?

A heavier mass will provide a large
restoring force, but it also needs a larger
force to achieve the same acceleration.

Because the acceleration is the same, the
periods will be the same.

Similarly, the larger amplitude requires a
larger restoring force. The acceleration will
be greater but the distance to move is also
greater. The period would stay the same.
Sample Problem 12B
Practice Problems

What is the period of a 3.98 m long
pendulum? What is the period of a 99.4 cm
long pendulum?
Periods of a mass-spring
system

Period of a mass-spring system depends on mass
and spring constant.
 A heavier mass attached to a spring increases
inertia without providing a compensating increase
in restoring force
 A heavy mass has a smaller acceleration than a
light mass has. So, a heavy mass has a greater
period.
 As mass increases, so does the period.

The greater the spring constant (k), the
stiffer the spring.
 A stiffer spring will take less time to
complete one cycle of motion than one that
is less stiff.
 As with the pendulum, changing the
amplitude of the vibration does not affect
the period.
Period of a mass-spring system in
simple harmonic motion

T = 2πm
k
Period=2π X square root of (mass/spring
constant)
Sample Problem 12C
Practice Problem 12C

A 1.0 kg mass attached to one end of a
spring completes one oscillation every 2.0 s.
Find the spring constant.
Conceptual Challenge

Why is a pendulum a reliable time-keeping
device, even if its oscillations gradually
decrease in amplitude over time?
12-3: Wave Motion






A wave motion travels away from the disturbance
that causes the wave.
Particles in the medium vibrate up and down as
the wave passes.
Ex: a leaf in a pond wave
Almost all wave types need a medium to travel
through, these are called mechanical waves.
Electromagnetic waves do not need a medium and
can travel in outer space
Ex: x-rays, microwaves, etc.
Wave Types

Pulse waves-a single traveling pulse
 Ex: Fig. 12-11; a single flip of the wrist
while holding one end of a rope that is fixed
 Periodic wave-continuous pulses that form
periodic motion such as moving your hand
up and down repeatedly while holding a
rope

Fig. 12-12
 As the sine wave created by this vibrating
blade travels to the right, a single point on
the string vibrates up and down with simple
harmonic motion

Transverse Waves-particles of the medium
vibrate perpendicularly to the motion of the
wave
 Fig. 12-13 waveform-represents the
displacements at a moment in time or the
displacements of a single particle as time
passes

Crest-the highest point above equilibrium
 Trough-lowest point below equilibrium
 Amplitude-maximum displacement from
equilibrium
 Fig. 12-13b
 A wave is a cyclical motion, first displaced
in one direction, then in the other direction,
then returning to equilibrium.

Wavelength-the distance the wave travels
during one cycle, λ
 Crest to crest=wavelength
 Trough to trough=wavelength





If a spring is pumped back and forth toward the
opposite fixed end, a longitudinal wave is formed
Vibrations are parallel to the motion of the wave
The spring should have compressed and stretched
regions of the coil that travel along the spring
Ex: sound waves
Fig. 12-15 the compressed regions correspond to
the crests and the stretched regions correspond to
troughs

Compressed regions are high density and
high pressure
 Stretched regions are low density and
pressure
Frequency, Period & Wave
Speed

Frequency of the vibrations of the source of a
wave equal the vibrations of the particles in the
wave
 Wave frequency describes the number of crests or
troughs that pass a given point in a unit of time
 Period of a wave describes the time it takes for a
complete wavelength to pass a given point
 Period is inversely related to frequency

A displacement of one wavelength occurs in a
time interval equal to one period of the vibration
 V=λ/T
 Substitute the inverse relationship, f=1/T
Into this equation gives us
V=fλ where the speed of a wave is constant except
when traveling from one medium to another
Sample Problem 12D

Waves carry energy as they move across a medium
while the medium doesn’t move
 Waves transfer energy by transferring motion of
matter rather than by transferring matter itself,
they do so efficiently
 The greater the amplitude, the more energy carried
in a given time
 The energy transferred is proportional to the
square of the wave’s amplitude

Ex: if amplitude of a wave is doubled, the
energy is increased by a factor of four
 If the amplitude is halved, the energy is
decreased by a factor of four
12-4: Wave Interactions

Two mechanical waves can occupy the
same space at the same time because they
are only displacements of matter, not matter
 When two waves pass through one another,
they form interference patterns of light and
dark bands called superposition, fig. 12-17
 Ex: electromagnetic radiation waves may
also interfere
Demo 10

Wave superposition
Constructive Interference

When two waves are traveling toward each
other with the same direction
displacements, they form a resultant wave
that is equal to the sum of the individual
waves, called superposition principle.

If the waves are on the same side of the
equilibrium, it is called constructive
interference.

Once the waves pass through each other,
their individual displacements are equal to
their previous displacements

Fig. 12-18
Destructive Interference

When the displacements are in opposite
directions from the equilibrium, the
resultant wave is the difference between the
pulses, called destructive interference

Fig. 12-19
Complete Destructive
Interference


When two pulses have equal but opposite
amplitudes, the resultant wave has a
displacement of zero, called complete
destructive interference.
Fig. 12-20
 After the interference, each pulse resumes
its previous amplitude

The superposition principle is true for
longitudinal waves as well
 A compression is a force on a particle in on
direction, while a rarefaction involves a
force on the same particle in the opposite
direction.
Reflection

At a free boundary, waves are reflected due
to an upward force at the boundary.
 Fig. 12-21 a

At a fixed boundary, waves are reflected
and inverted due to a downward force at the
boundary.
 Fig. 12-21b
Standing Waves

When a string is shaken up and down in a
regular motion, it produces waves with the
same frequency, wavelength, and amplitude.
Those waves are then sent down the string
and reflected back down the string toward
each other.
 The result is a standing wave-a wave pattern
that does not move down the string.

A standing wave contains both constructive
and destructive interference
 Fig. 12-22a
 Nodes-points at which two waves cancel,
complete destructive interference
 Antinodes-point at which the largest
amplitude occurs, constructive interference
Fig. 12-23

Waves running together create a blur pattern
 A single loop represents a crest or trough
alone while two loops correspond to a crest
and a trough together, one wavelength
 The ends of the strings must be nodes, thus
it is a standing wave
Fig. 12-23 Standing
Waves/Wavelength

The wavelengths of these standing waves
depends on the string lengths
 Wavelengths:
 A. String length, L
 B. 2L
 C. L
 D. 2/3L
Demo 10-Wave Superposition

Two students hold long coiled spring
 One student send a single pulse down the
spring
 The opposite student then sends an identical
pulse.
 Both students generate pulses
simultaneously.

Where do the pulses cross each other?
 How can we tell?
 How can we tell that the pulses are crossing
through each other and not bouncing off
each other?
 Now, send two pulses of very different
amplitudes toward each other.
Demo 11-Waves passing each
other

Same scenario as demo 10
 Create pulses with opposite displacements
and observe the waves that reach the hands
of the students
 Observe the same using waves with
different amplitudes and displacements on
the same side.
 Observe constructive and destructive
Demo 12-Wave Reflection

Fix one each of a spring to an object and
hold the other in your hand.
 Send a pulse down the spring and observe
the reflected pulse.
 Is the pulse inverted?
 Try the same with a piece of rope fixed and
then loosely tied to an object.
Demo 13-Standing Waves

Create a standing wave with the springs