Transcript Chapter 13

Chapter 11
Vibrations and Waves
Ms. Hanan
11-1 Simple Harmonic Motion
Objectives
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Identify the conditions of simple harmonic
motion.
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Explain how force, velocity, and acceleration
change as an object vibrates with simple
harmonic motion.
•
Calculate the spring force using Hooke’s law.
Vocabulary
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Periodic Motion
Simple Harmonic Motion
Period
Amplitude
Hooke’s Law
Pendulum
Oscillation
Vibration
Spring Constant
Displacement
Periodic Motion
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Motion where a body travels along the same path in a
repeated, back and forth manner
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Also called oscillation and vibration
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We are surrounded by oscillations – motions that repeat
themselves (periodic motion)
• Grandfather clock pendulum, boats bobbing at anchor,
oscillating guitar strings, pistons in car engines
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Understanding periodic motion is essential for the study of
waves, sound, alternating electric currents, light, etc.
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An object in periodic motion experiences restoring forces
or torques that bring it back toward an equilibrium position
Periodic Motion
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Those same forces cause the object to “overshoot”
the equilibrium position
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Think of a block oscillating on a spring or a pendulum
swinging back and forth past its equilibrium position
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Examples of periodic motion:
R
L
/5
m
r
L
/2
k
m
Example 1
Mass-Spring System
a
a
a
Equil. position
a
Example 2
Simple Pendulum
a
a
Equil. position
a
a
Example 3
Floating Cylinder
Equil. position
a
a
a
a
Hooke’s Law Force
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LEQ
k
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x
m
The force always
acts toward the
equilibrium
position
The direction of
the restoring
force is such
that the object
Fs=kx is being either
pushed or pulled
toward the
equilibrium
position
Hooke’s Law Reviewed
F  kx
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When x is positive
F is negative
;
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When at equilibrium (x=0),
F = 0 ;
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When x is negative
F is positive
;
,
,
Stretched and Equilibrium
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Equilibrium and Compressed
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Motion of the Spring-Mass System
• Assume the object is initially pulled to a
distance A and released from rest
• As the object moves toward the
equilibrium position, F and a decrease,
but v increases
• At x = 0, F and a are zero, but v is a
maximum
• The object’s momentum causes it to
overshoot the equilibrium position
Graphing x vs. t
A
T
A : amplitude (length, m)
T : period (time, s)
Sample Problem A – P. 370
If a mass of 0.55 kg attached to a vertical
spring stretches the spring 2.0 cm from its
equilibrium position, what is the spring
constant?
Givens:
m = 0.55 kg
x = -2.0 cm
x = -0.02 m
g = 9.81 m/s2
Unknowns:
k = ?
X = -2.0 cm
Step 1: Choose the equation/situation:
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When the mass is attached to the spring, the equilibrium
position changes.
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At the new equilibrium point, the net force acting on the mass
is zero.
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By Hooke’s Law, the Spring Force must be equal and opposite
of the weight of the mass
FSpring  kx
Fg  mg
Fnet  FSpring  Fg  0
(kx)  (mg)  0
kx  m g
 mg
k
x
 mg
k
x
Step 2: Substitute the known values into this equation.
m = 0.55 kg
x = -0.02 m
g = 9.81 m/s2
 (0.55kg )(9.81m / s )
k
 0.02m
2
k  270 N / m
Assignments
• Class-work:
Practice A , page 371, questions 1,
2, and 3.
• Homework:
Review and Assess; Page 396: # 8
and 9
Elastic Potential Energy
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The energy stored in a stretched or
compressed spring or other elastic material
is called elastic potential energy
PEs = ½kx2
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The energy is stored only when the spring is
stretched or compressed
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Elastic potential energy can be added to the
statements of Conservation of Energy and
Work-Energy
Energy Transformations
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The block is moving on a frictionless surface
The total mechanical energy of the system is
the kinetic energy of the block
Energy Transformations, 2
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The spring is partially compressed
The energy is shared between kinetic energy
and elastic potential energy
The total mechanical energy is the sum of the
kinetic energy and the elastic potential energy
Energy Transformations, 3
• The spring is now fully compressed
• The block momentarily stops
• The total mechanical energy is stored as
elastic potential energy of the spring
Energy Transformations, 4
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When the block leaves the spring, the total
mechanical energy is in the kinetic energy of
the block
The total energy of the system remains
constant
Simple Pendulum
Restoring force of a pendulum is a
Component of the bob’s weight
x 2  L2
x
F  mgsin 
x
x
sin  

x 2  L2 L
mg
F
x
L
Looks like Hooke’s law (k  mg/L)
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When oscillations are small, the motion is called
simple harmonic motion (shm) and can be described
by a simple sine curve.
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The pendulum’s potential energy is gravitational, and
increases as the pendulum’s displacement increases.
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Gravitational potential energy is equal to zero at
the pendulum’s equilibrium position. PEg = mgh
Assignments
• Class-work:
Practice section review page 375,
questions 1, 2, 3, and 4.
• Homework:
Vibrations and Waves Problem A,
Hooke’s Law Additional Practice
Sheet, even questions. Due Sunday
20/2/11
11-2 Measuring Simple harmonic Motion
Objectives
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Identify the amplitude of vibration.
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Recognize the relationship between period and
frequency.
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Calculate the period and frequency of an
object vibrating with simple harmonic motion.
Amplitude, Period, and Frequency
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Amplitude is the maximum displacement from
equilibrium.
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Period is the time it takes to execute a complete
cycle of motion.
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Frequency is the number of cycles or vibrations
per unit of time.
Amplitude
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Pendulum: amplitude can be measured by the angle
between the pendulum’s equilibrium position and its
maximum displacement.
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Mass-spring system: amplitude is the maximum
amount the spring is stretched or compressed from
its equilibrium position.
Period and frequency measure time
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Swinging from maximum
displacement on one side of
equilibrium to maximum
displacement on the other side
and back again = one cycle
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Period (T): the time it takes
for this complete cycle of
motion. Units: second, s
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Frequency (f): the number of
complete cycles in a unit of
time. Units: 1 s-1= 1 Hz
Period of a Simple Pendulum in Simple
Harmonic Motion
Depends on string length and free-fall acceleration.
L = length
Sample Problem B
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 its period is 12s. How
tall is the tower?
Given:
T = 12 s
Unknown:
L = ?
g = 9.81 m/s2
Sample Problem 12B
L
T  2
g
L
12s  2
9.81 sm2
4 L
144s 
9.81 sm2
2
2
144s 9.81   L
2
4
L  36m
m
s2
2
Assignments
• Class-work:
Practice B , page 379, questions 1,
2, 3, and 4.
• Homework:
Section review on page 375 odd
questions
Review; Page 397: # 19 and 20
Period of a Mass-Spring System
Depends on mass and spring constant.
Sample Problem C
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.
Given:
T = 0.840 s
Unknown:
k=?

1275kg 153kg
m
 357kg
4
Sample Problem B
T  2
m
k
m
T  4  
k
4 2 m 4 2 357kg 
k

2
T
0.840s 2
2
2
k  2.00  104 N / m
Assignments
• Class-work:
Practice c , page 381, questions 1,
2, 3, 4, and 5.
• Homework:
Section review on page 381, odd
questions
Review; Page 397: # 21
11-3 Properties of Waves
Objectives
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Distinguish local particle vibrations from overall
wave motion.
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Differentiate between pulse waves and periodic
waves.
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Interpret waveforms of transverse and
longitudinal waves.
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Apply the relationship among wave speed,
frequency, and wave length to solve problems.
Vocabulary
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Wave
Mechanical wave
Medium
Transverse wave
Crest
Trough
Wavelength
Longitudinal wave
Wave Motion
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The process by which a disturbance at one point
is propagated to another point more remote from
the source with no net transport of the material
of the medium itself;
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Examples: the motion of electromagnetic waves,
sound waves, hydrodynamic waves in liquids, and
vibration waves in solids.
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Medium–material through which a disturbance
travels.
Types of Waves
• Mechanical Wave–a wave whose
propagation requires the existence of a
medium.
• Electromagnetic Waves–a wave consisting
of oscillating electric and magnetic fields
at right angles to each other, no medium
is required.
Kinds of Waves
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Transverse wave–A wave in which the vibration
is at right angles (perpendicular) to the
direction in which the wave is traveling.
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Longitudinal Wave–A wave in which the
vibration is in the same direction (parallel) as
that in which the wave is traveling.
Transverse and Longitudinal Waves
Single or Multiple
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Wave Pulse–A single disturbance traveling
through a medium.
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Periodic Wave–A wave whose source is some
form of periodic motion.
Parts of Waves
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Crests–One of the places in a wave where the
wave is highest or the disturbance is greatest.
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Troughs–One of the places in a wave where the
wave is lowest or the disturbance is greatest in
the opposite direction from a crest.
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Wavelength–The distance from the top of crest
of a wave to the top of the following crest, or
equivalently, the distance between successive
identical parts of the wave.
Parts of Waves 2
Wave Speed
x
v
t
v

T
Velocity
For waves
x  
t  T

1
f
T

v   f
T
Speed of a Wave
v  f
Longitudinal (Compression) Waves
Sound waves are longitudinal waves
Transverse Waves
Elements move perpendicular to wave motion
Elements move parallel to wave motion
VIBRATION OF A PENDULUM
What does the period (T) depend upon?
Length of the pendulum (L).
Acceleration due to gravity (g).
Period does not depend upon the bob’s
mass or the amplitude of the swing.
T  2 l g
Vibration of a pendulum.
The to-and-from vibratory
motion is also called
oscillatory motion (or
oscillation).
Damped Oscillations
In real systems,
friction slows motion
Assignments
• Class-work:
Practice D , page 387, questions 1,
2, 3, and 4.
• Homework:
Section review on page 388 odd
questions.
Review; Page 397: # 34 and 35