AP Physics
Chapter 13
Vibrations and Waves
Chapter 13: Vibrations and Waves
13.1
13.2
13.3
13.4
13.5
Simple Harmonic Motion
Equations of Motion
Wave Motion
Wave Properties
Standing Waves and Resonance
Homework for Chapter 13
• Read Chapter 13
• HW 13.A : pp.447-448: 8,9,11,12,15,16,17,20,31-38, 41,42.
• HW 13.B: pp 449-450: 51-55, 58,60,62,64,71,76-80.
13.1: Simple Harmonic Motion
Warmup: Good Vibrations
Physics Warmup #114
The source of all waves is a vibrating object.
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Complete the table below by identifying the source of each wave described.
Wave
Source
A tuning fork is struck with a rubber
hammer, producing a sound wave.
the vibrating tuning fork
A motorboat moves through the
water, leaving its wake behind.
the propeller blades
A performer sings a high note.
A light bulb gives off light.
vocal cords
vibrating electrons
13.1: Simple Harmonic Motion
• The motion of an oscillating object depends on the restoring forces that make it
go back and forth.
• The simplest type of restoring force is a spring force.
Hooke’s Law:
Fs = -kx
where k is the spring constant and
x is the displacement
• The negative sign indicates that the force is opposite to the
displacement from the springs relaxed position.
• Motion under the influence of the type of force described by
Hooke’s Law is called:
simple harmonic motion (SHM)
• It is called harmonic because the motion can be described by sines
and cosines.
13.1: Simple Harmonic Motion
Fig 13.1, p. 420.
A block on a spring undergoes simple harmonic motion.
a) The block is at the equilibrium position, x = 0.
a) The force of a hand, Fh, pulls the block for a displacement
of x = A. The force of the spring is Fs.
• At the time of the release, t = 0.
• The time it takes to complete one period of oscillation is T.
c) At t = T/4, the block is back at the equilibrium position.
d) at t = T/2, the block is at x = -A.
e) During the next half of the cycle, the motion is to the right.
f) At t = T, the object is back at its starting position.
13.1: Simple Harmonic Motion
displacement
- the distance of an object, including direction (  x),
from its equilibrium position.
amplitude (A)
- the magnitude of the maximum displacement of a
mass from its equilibrium position.
period (T)
- the time needed to complete one cycle of oscillation.
frequency (f)
- the number of cycles per second.
• frequency and period are related by:
f= 1
T
• The SI unit of frequency is 1/s, or hertz (Hz). This is also known as cycles per
second.
13.1: Simple Harmonic Motion
The Energy and Speed of a Spring-Mass System in SHM
Recall from Chapter 5, the total potential energy stored in a spring is:
U = ½ kx2
On Gold Sheet
The total kinetic and potential energies of a spring-mass system is equal to its total
mechanical energy.
E = K + U = ½ mv2 + ½ kx2
At a point of maximum displacement, (-A or +A), the instantaneous velocity is zero.
Therefore all the energy at this point is potential.
E = ½ m(0)2 + ½ k( A) 2
Simplifying, the total energy in SHM of a spring:
E = ½ kA2
**Energy
is proportional to the square
of the amplitude**
13.1: Simple Harmonic Motion
Example 13.1: A 0.50 kg object is attached to a spring of spring constant 20
N/m along a horizontal frictionless surface. The object oscillates in simple
harmonic motion and has a speed of 1.5 m/s at the equilibrium position.
a) What is the total energy of the system?
b) What is the amplitude?
c) At what location are the values for the potential and kinetic energies the
same?
13.1: Simple Harmonic Motion
Example 13.2: An object is attached to a spring of spring constant 60 N/m
along a horizontal, frictionless surface. The spring is initially stretched by a
force of 5.0 N on the object and let go. It takes the object 0.50 s to get back
to its equilibrium position after its release.
a) What is the amplitude?
b) What is the period?
c) What is the frequency?
13.1: Simple Harmonic Motion: Check for Understanding
1. A particle in SHM:
a. has variable amplitude
b. has a restoring force in the form of Hooke’s Law
c. has a frequency directly proportional to its period
d. has its position represented graphically by x(t) = at + b
Answer: b
13.1: Simple Harmonic Motion: Check for Understanding
2. The maximum kinetic energy of a spring-mass system in SHM is equal to:
a. A
b. A2
c. kA
d. kA2/2
Answer: d
13.1: Simple Harmonic Motion: Check for Understanding
3. If the amplitude of an object in SHM is doubled:
a. how is the energy affected?
b. how is the maximum speed affected?
Answer:
a. Since E = ½kA2, the energy is four times as large.
b. Since vmax =
k
m
A , the maximum speed is twice as large.
13.1: Simple Harmonic Motion: Check for Understanding
4. If the period of a system in SHM is doubled, its frequency is:
a. doubled
b. halved
c. four times as large
d. one-quarter as large
Answer: b, because f = 1/T
13.1: Simple Harmonic Motion: Check for Understanding
5. When a particle in SHM is at the equilibrium position, the potential energy of the
system is:
a. zero
b. maximum
c. negative
d. none of the above
Answer: a, because U = ½ kx2
13.2: Equations of Motion
Warmup: Famous Scientists II
Physics Warmup #153
Men and women are still making discoveries that totally change our ideas about
certain areas of science, revise our theories, and in some cases, abandon
centuries-old explanations. Most of those making the discoveries had no idea
where technology would take their new found knowledge. Such was the case with
Ernest Rutherford, Neils Bohr, and Enrico Fermi and their contributions toward our
understanding of the atom.
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Solve this anagram to identify a famous scientist not mentioned.
sent elite brain
Answer: Albert Einstein
13.2: Equations of Motion
Simple Harmonic Motion can be defined using a reference circle as follows:
“If a particle is undergoing uniform circular motion then its projection on any
diameter of its circular path performs Simple Harmonic Motion.”
View the animation:
http://137.229.52.100/physics/p103/applets/ref_circle.html
For this chapter, we will be using radians. Make sure to adjust your calculator!
 is measured in radians (rad)
 is measured in radians/second (rad/sec)
A is the radius of the reference circle.
13.2: Equations of Motion
The reference circle for horizontal motion
a. The shadow of an object in uniform circular motion has the same horizontal
motion as the object on a spring in SHM.
b. The motion equation can be written x = A cos Ɵ or x = A cos t.
13.2: Equations of Motion
The reference circle for vertical motion
a. The shadow of an object in uniform circular motion has the same vertical motion
as the object oscillating on a spring in SHM.
a. The motion equation can be written as y = A sin Ɵ or y = A sin t.
13.2: Equations of Motion
equation of motion
- gives the object’s position as a function of
time.
ex: for constant acceleration, we use kinematics formulas,
such as x = vo + at. Simple harmonic motion does NOT have constant
acceleration, so we can’t use kinematics equations.
•The equations of motion for an object in SHM is a combination of simple
harmonic and uniform circular motion. They are:
Recall,
y = A sin (t + )
where y is the vertical displacement (in meters)
A is the amplitude (in meters)
 is the angular frequency of motion (in rad/sec)
 is the phase constant (in rad)
 = 2f = 2
T
Remember to set your calculator to radians!
•  is the phase constant. It is determined by the initial displacement and
velocity direction. It will help you decide whether to use the sine or cosine
function to describe a particular case of SHM.
The phase difference
between sine and cosine
is 90° or /2.
13.2: Equations of Motion
a) If y=0 at t=0, and the
motion is initially
upward, the curve
corresponds with a
sine wave.
b) If the initial condition
has positive amplitude,
the wave, the curve
corresponds with a
cosine wave.
c) Here, the motion is
initially downward and
y = 0 at t = 0. A is
negative; it is a sine
wave.
d) Finally, the initial
amplitude negative; it
is a cosine wave.
13.2: Equations of Motion
Observe how sinusoidal curve is traced out on the moving paper.
Since the object’s initial displacement is +A, the equation can be written as
y = A cos t
13.2: Equations of Motion
Two other equations of motion for an object in SHM are:
velocity:
v =  A cos (t + )
acceleration:
a = -2 A sin (t + ) = - 2 y
13.2: Equations of Motion
a) The mass is held, then
released.
b) The weight of the mass
makes it drop.
c) The restoring force of the
spring pulls back.
d)The mass is in SHM.
• Velocity is /2 out of phase
with displacement.
• Acceleration is  out of phase
with displacement.
13.2: Equations of Motion
damped harmonic motion
- without a driving force, the amplitude
or energy of an oscillating body will decrease with time.
1.00
displacement (m)
0.75
0.50
0.25
0.00
-0.25
-0.50
-0.75
-1.00
time
View simulation:
http://physics.bu.edu/~duffy/semester1/c19_damped_sim.html
13.2: Equations of Motion
T=2
m
k
Period of an object oscillating on a spring
The frequency f of the oscillation is equal to 1/T. Therefore,
f= 1 k
2 m
or  =
Frequency of an object oscillating on a spring
k
m
• Note that the time period depends on the mass of the object and the
spring constant, but does not depend on the acceleration due to gravity.
• The greater the mass, the longer the period. The greater the spring
constant (the stiffer the spring), the shorter the period.
13.2: Equations of Motion
A simple pendulum
heavy object on a string.
consists of a small,
For small angles of oscillation (Ɵ < 10°),
a good approximation for period is:
T=2
L
g
Period of a simple pendulum On Gold Sheet
where L is the length of the string
g is the acceleration due to gravity
• Note that the period of the simple pendulum is independent of the
mass of the bob and the amplitude of the oscillations.
13.2: Equations of Motion
Example 13.3: An object with a mass of 1.0 kg is attached to a spring with a
spring constant of 10 N/m. The object is displaces by 3.0 cm from the
equilibrium position and let go.
a) What is the amplitude A?
b) What is the period T?
c) What is the frequency f?
13.2: Equations of Motion
Example 13.4: The pendulum of a grandfather clock is 1.0 m long.
a) What is its period on the Earth?
b) What would its period be on the Moon where the acceleration due to
gravity is 1.7 m/s2?
13.2: Equations of Motion
Example 13.5: The position of an object in simple harmonic motion is
described by y = (0.25 m) sin (/2 t). Find the
a) amplitude A
b) period T
c) maximum speed
13.2: Equations of Motion: Check for Understanding
1. The equation of motion for a particle in SHM
a. is always a cosine function
b. reflects damping action
c. is independent of the initial conditions
d. gives the position of the particle as a function of time
Answer: d
13.2: Equations of Motion: Check for Understanding
2. If the length of a pendulum is doubled, what is the ratio of the new
period to the old one?
a. 2
b. 4
c. 1/2
d. 2
Answer: d
13.2: Equations of Motion: Check for Understanding
3. Which of the following does not affect the period of a vibrating mass
on a spring?
a. mass
b. spring constant
c. acceleration due to gravity
d. frequency
Answer: c
Homework for Chapter 13.1 & 13.2
• HW 13.A: pp.447-448: 8,9,11,12,15,16,17,20,31-38, 41,42.
13.3: Wave Motion
Warmup: Type Casting
Physics Warmup #116
Waves that cause a medium to be disturbed in a direction perpendicular to the
direction in which the wave is traveling are called transverse waves. When the
medium is disturbed in a direction parallel to the direction in which the wave is
traveling, the wave is called longitudinal.
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Complete the table below by identifying each wave as being either transverse or
longitudinal.
Wave
Source
A sound wave
longitudinal
A water wave caused by a boat moving
transverse
A wave in a rope caused by one end
being moved up and down
transverse
A wave in a coiled spring caused by
pushing one end in and out repeatedly
longitudinal
A light wave
transverse
13.3: Wave Motion
13.3: Wave Motion
wave motion
- the propagation of a disturbance (energy and
momentum) through a material.
• Only energy is transferred, not matter.
periodic wave
- requires a disturbance from an oscillation source.
• If the driving source maintains constant amplitude of the wave, the
result is SHM.
A periodic wave can be characterized by the following:
amplitude
- the magnitude of displacement of the particles of the
material from their equilibrium position.
wavelength
- the distance between two successive crests or troughs.
frequency
in a second.
- the number of wavelengths that passes by a given point
wave speed
- the speed of wave motion (speed of a crest or trough)
given by:
v =  f = /T
where  is wavelength, f is frequency, and T is period
13.3: Wave Motion
The rope “particles” oscillate vertically in simple harmonic motion.
The distance between two successive points that are in phase (at identical
points on the wave form) is the wavelength.
Question: What is the phase difference between the first (red) and last (blue)
waves?
Answer: /2
13.3: Wave Motion
Example 13.6: A student reading her physics book on a lake dock notices that
the distance between two incoming wave crests is about 2.4 m, and she then
measures the time of arrival between wave crests to be 1.6 s. What is the
approximate speed of the waves?
Answer:
 = 2.4 m
T = 1.6 s
v = /T = 2.4 m / 1.6 s
v = 1.5 m/s
13.3: Wave Motion
13.3: Wave Motion
transverse wave
- the particle
motion is perpendicular to the direction of the
wave velocity.
ex: guitar string; electromagnetic
wave
longitudinal wave
- the particle
oscillation is parallel to the direction of the
wave velocity.
• also called a compressional wave
• can propagate in solids, liquids, or
gases
ex: sound waves
Combination of transverse and longitudinal
waves: ex: seismic, water
View Wave Motion:
http://paws.kettering.edu/~drussell/Demos/w
aves/wavemotion.html
Rarefaction
13.3: Wave Motion: Check for Understanding
13.3: Wave Motion: Check for Understanding
13.3: Wave Motion: Check for Understanding
13.3: Wave Motion: Check for Understanding
13.3: Wave Motion: Check for Understanding
13.3: Wave Motion: Check for Understanding
13.4: Wave Properties
13.4: Wave Phenomena
interference
- when waves meet or overlap
principle of superposition
- at any time, the waveform of two or
more interfering waves is given by the sum of the displacements of the individual
waves at each point in the medium.
constructive interference
- if the amplitude of the combined
wave is greater than that of any of the individual waves.
destructive interference
- if the amplitude of the combined wave
is smaller than that of any of the individual waves.
Create your own interference:
http://id.mind.net/~zona/mstm/physics/waves/interference/waveInterference1/
WaveInterference1.html
13.4: Wave Phenomena
total constructitve interference
- when two waves of the same frequency and
amplitude are exactly in phase.
ex: the crest of one wave is aligned with the crest of another.
total destructive interference
- when two waves of the same frequency and
amplitude are completely 180° out of phase.
ex: the crest of one wave meets the trough of another.
13.4: Wave Phenomena
reflection
- when a wave strikes
and object or comes to a boundary of another
medium and is at least partly bounced back.
ex: an echo is a reflected sound wave
ex: a mirror reflects light waves
a) When a wave is reflected from a fixed boundary,
the reflected wave is inverted, or undergoes a
180° phase shift.
b) If the string is free to move at the boundary, there
is no phase shift of the reflected wave.
13.4: Wave Phenomena
refraction
- when a wave crosses a boundary into another medium
and the transmitted wave moves in a different direction.
• When a wave crosses a boundary into another medium, its speed changes.
• When the incident wave enters at an angle, the transmitted wave moves in a
different direction.
• Generally, when a wave strikes the boundary, both reflection and refraction
occur.
a) Refraction of water waves.
As the crests approach the beach, their left
edge slows as it enters the shallow water
first. The whole crest rotates, approaching
the beach more or less head on.
13.4: Wave Phenomena
dispersion
- waves of different frequencies spread apart form one another.
• Nondispersive waves travel at the same speed which is determined solely by the
properties of the medium. They do not depend on the wavelength (or frequency) or
the wave.
ex: a wave on a string, sound
• Dispersion examples: light in water, rainbows
b) Surface wave traveling in deep water are dispersive. The wave speed depends on
frequency or wavelength.
13.4: Wave Phenomena
diffraction
- the bending of waves around and object.
ex: A person in a room with an open door can hear sound from outside
the room, partially due to diffraction.
Diffraction Applets:
http://www.ngsir.netfirms.com/englishhtm/Diffraction.htm
http://www.ngsir.netfirms.com/englishhtm/Diffraction3.htm
13.4: Wave Phenomena: Check for Understanding
1. When waves meet each other and iinterfere, the resultant waveform is
determined by
a) reflection
b) refraction
c) diffraction
d) superposition
Answer: d
13.4: Wave Phenomena: Check for Understanding
2. Refraction
a) involves constructive interference
b) refers to a change in direction at the media interfaces
c) is synonymous with diffraction
d) occurs only for mechanical waves, not light
Answer: b
13.4: Wave Phenomena: Check for Understanding
13.4: Wave Phenomena: Check for Understanding
13.5: Standing Waves and
Resonance
Warmup: Foot Stompin’
Physics Warmup #117
Resonance occurs when the frequency of a forced vibration on an object matches
the object’s natural frequency. This causes a great increase in amplitude, which
increases the power transmitted by the object. In 1940, the Tacoma Narrows
suspension bridge collapsed when wind-driven oscillations produced resonance in
the bridge. Films of its collapse have become favorites among physics teachers an
their students. Subsequent designs have incorporated such innovations as
separate parallel roadways as a way to keep this type of disaster from happening
again.
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In the 1800’s, English soldiers marching across a small suspension bridge caused
it to collapse when their marching set it into resonance. Their marching was in
rhythm with the bridge’s natural frequency. Since that time, soldiers and marching
bands have been told to not march in step when crossing any type of suspension
bridge.
Give another example of the disastrous effects of resonance and describe how it
happens.
13.5: Standing Waves and Resonance
standing wave
- occur when interfering waves of the same frequency
and amplitude traveling in opposite directions, such as in a rope
• Standing waves can be generated in a rope by more than one driving
frequency.
• The higher the frequency, the more “loops” in the rope.
nodes
- the points on the rope that are always stationary due to
destructive interference
• adjacent nodes are separated by a half wavelength, or 
antinodes
- the points of maximum amplitude, where constructive
interference is greatest
• adjacent antinodes are separated by a half wavelength, or 
13.5: Standing Waves and Resonance
natural frequencies
or resonant frequencies
which large-amplitude standing waves are produced.
- the frequencies at
• The resulting standing wave patterns are called normal, characteristic,or
resonant, modes of vibration.
• A stretched string or rope fixed at both ends can be analyzed to determine its
natural frequencies.
• The number of loops of a standing wave that will fit between the nodes at the ends
is equal to an integral number of half-wavelengths.
L = n n
2
or
n = 2L (for n = 1,2,3,…)
n
where n is the number of loops or half-wavelengths,
L is the length of string, and  is the wavelength
• A stretched string can have standing waves only at certain frequencies. These
frequencies correspond to the number of half-wavelength loops that will fit on the
string between the ends.
13.5: Standing Waves and Resonance
13.5: Standing Waves and Resonance
• The natural frequencies of oscillation for waves on a string are:
fn = v = n v
n
2L
= n f1 (for n = 1,2,3,…)
where v is the speed of waves on a string
fundamental frequency (f1)
- the lowest natural frequency
• All other natural frequencies are integral multiples of the fundamental frequency.
fn = n f1 ( for n = 1, 2,3,…)
harmonic series
- the set of frequencies f1, f2=2f1, f3=3f1,…,
• f1, the fundamental frequency, is called the first harmonic
• f2, is called the second harmonic, etc.
Experiment with Harmonics on a Musical Instrument:
http://id.mind.net/~zona/mstm/physics/waves/standingWaves/standingWaves1/Sta
ndingWaves1.html
13.5: Standing Waves and Resonance
Natural frequencies
also depend on factors
such as the mass of
the string and the
tension.
13.5: Standing Waves and Resonance
• The natural frequencies of a stretched string can also be written as:
fn = n v
2L
= n
2L
FT

= n f1
(for n = 1,2,3,…)
• Note, the greater the linear mass density of a string, the lower its natural
frequencies.
ex: the low note strings on a guitar are thicker, or more massive, than
the higher note strings
ex: tightening a string increases all the frequencies of that string
13.5: Standing Waves and Resonance
Example 13.7: A 50 m long string has a mass of 0.010 kg. A 2.0 m segment of
the string is fixed at both ends and when a tension of 20 N is applied to the
string, three loops are produced. What is the frequency of the standing wave?
13.5: Standing Waves and Resonance
resonance
- when the driving frequency of an external source
matches a natural frequency of the system.
• the vibrational amplitude is greatly enhanced
Pushing a person in a swing is a common example of resonance. The loaded
swing, a pendulum, has a natural frequency of oscillation, its resonant frequency,
and resists being pushed at a faster or slower rate.
Resonance can be desirable and undesirable. Read Insight, p. 445.
13.5: Standing Waves and Resonance: Check for Understanding
1. For two traveling waves to form standing waves, the waves must have the
same
a) frequency
b) amplitude
c) speed
d) all of the preceding
Answer: d
13.5: Standing Waves and Resonance: Check for Understanding
2. When a stretched violin string oscillates in its third harmonic mode, then the
standing wave in the string will exhibit
a) 3 wavelengths
b) 1/3 wavelength
c) 3/2 wavelengths
d) 2 wavelengths
Answer: c
13.5: Standing Waves and Resonance: Check for Understanding
13.5: Standing Waves and Resonance
Homework for Chapter 13.3, 13.4,
& 13.5
• HW 13.B: pp 449-450: 51-55, 58,60,62,64,71,76-80.