Download Oscillations and Waves

Chapter 14
Periodic Motion
PowerPoint® Lectures for
University Physics, Thirteenth Edition
– Hugh D. Young and Roger A. Freedman
Lectures by Wayne Anderson
Copyright © 2012 Pearson Education Inc.
Goals for Chapter 14
• To describe oscillations in terms of amplitude, period,
frequency and angular frequency
• To do calculations with simple harmonic motion
• To analyze simple harmonic motion using energy
• To apply the ideas of simple harmonic motion to
different physical situations
• To analyze the motion of a simple pendulum
• To examine the characteristics of a physical pendulum
• To explore how oscillations die out
• To learn how a driving force can cause resonance
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What causes periodic motion?
• If a body attached to a spring
is displaced from its
equilibrium position, the
spring exerts a restoring force
on it, which tends to restore
the object to the equilibrium
position. This force causes
oscillation of the system, or
periodic motion.
• Figure 14.2 at the right
illustrates the restoring force
Fx.
Fx   kx
ma   kx
d 2x
k
a 2  x
dt
m
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Characteristics of periodic motion
• The amplitude, A, is the maximum magnitude of displacement
from equilibrium.
• The period, T, is the time for one cycle.
• The frequency, f, is the number of cycles per unit time.
• The angular frequency, , is 2π times the frequency:  = 2πf.
• The frequency and period are reciprocals of each other:
f = 1/T and T = 1/f.
• Follow Example 14.1.
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Simple harmonic motion (SHM)
• When the restoring force is directly proportional to the displacement from
equilibrium, the resulting motion is called simple harmonic motion (SHM).
• An ideal spring obeys Hooke’s law, so the restoring force is Fx = –kx, which
results in simple harmonic motion.
Fx   kx
ma   kx
d 2x
k
a 2  x
dt
m
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Simple harmonic motion viewed as a projection
• Simple harmonic motion is the projection of uniform circular
motion onto a diameter, as illustrated in Figure 14.5 below.
x  A cos  A cos t
d 2x
k


x SHM
dt 2
m
d 2x
k
2



A
cos

t


A cos t
dt 2
m
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2 
k
k
 
m
m
Characteristics of SHM
• For a body vibrating by an ideal spring:
k
 m
k T  1  2  2 m
f  1 m
2 2
f 
k
• Follow Example 14.2 and Figure 14.8 below.
(a) Find force constant k of the spring
(b) Find angular frequency, frequency, and period of oscillation
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Displacement as a function of time in SHM
• The displacement as a function of
time for SHM with phase angle 
is x = Acos(t + ). (See Figure
at right.)
• Changing m, A, or k changes the
graph of x versus t, as shown
below.
change
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
Displacement, Velocity and Acceleration
• The displacement as a function of time for SHM with phase angle  is:
x(t )  A cos(t   )
• As always, velocity is the time-derivative of displacement:
vx (t ) 
dx
  A sin(t   )
dt
• Likewise, acceleration is the time-derivative of velocity (or the second
derivative of displacement):
dvx d 2 x
ax (t ) 
 2   2 A cos(t   )
dt
dt
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Graphs of displacement, velocity, and acceleration
• The graph below
shows the effect of
different phase angles.
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• The graphs below show x, vx,
and ax for  = π/3.
Behavior of vx and ax during one cycle
• Figure 14.13 at the right shows
how vx and ax vary during one
cycle.
• Example 14.3: A glider attached
to a spring is given an initial
displacement 0.015 m and a push
with velocity +0.40 m/s.
(a) Find the period, amplitude and
phase of the resulting motion.
(b) Write equations for the
displacement, velocity and
acceleration as a function of
time.
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Energy in SHM
• The total mechanical energy E = K + U is conserved in SHM:
E = ½ mvx2 + ½ kx2 = ½ kA2 = constant
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Energy diagrams for SHM
• Figure 14.15 below shows energy diagrams for SHM.
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Energy and momentum in SHM
• Example 14.5:
• A block of mass M attached to a
horizontal spring with force constant k
is moving in SHM with amplitude A1.
(a) As the block passes through its
equilibrium position, a lump of putty
of mass m is dropped from a small
height and sticks to it. Find the new
amplitude and period of the motion.
(b) Repeat part (a) if the putty is
dropped onto the block when it is at
one end of its path.
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Angular SHM
• A coil spring (see Figure 14.19 below) exerts a restoring torque
z = –, where  is called the torsion constant of the spring.
• The result is angular simple harmonic motion.
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The simple pendulum
• Other systems can show SHM.
• Consider a simple pendulum that
consists of a point mass (the bob)
suspended by a massless,
unstretchable string.
• If the pendulum swings with a
small amplitude  with the
vertical, its motion is simple
harmonic, where the restoring
force is the component of gravity
along the arc of the motion.
F ( )  ma  mg sin 
d 2
ml 2  mg
dt
d 2
g



dt 2
l
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
g
; 2 f 
l
g
1
l
; T   2
l
f
g
The physical pendulum
• A physical pendulum is any
pendulum that uses an extended
body instead of a point-mass
bob.
• For small amplitudes, its
motion is simple harmonic.
(See Figure 14.23 at the right.)

mgd
2
I
; T
 2
I

mgd
• Example 14.9: A uniform rod
of length L is pivoted about one
end, what is its period of
oscillation?
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Damped oscillations
•
Real-world systems have some
dissipative forces that decrease the
amplitude.
•
Such dissipative forces are typically
proportional to the speed v, and
appear in the force equation with
minus sign:
d 2x
dx
Fx  kx  bvx  m 2  kx  b
dt
dt
• The decrease in amplitude is called
damping and the motion is called
damped oscillation.
•
The mechanical energy of a damped
oscillator decreases continuously.
The general solution is:
x  Ae(b /2m)t cos(t   )
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Chapter 15
Mechanical Waves
PowerPoint® Lectures for
University Physics, Thirteenth Edition
– Hugh D. Young and Roger A. Freedman
Lectures by Wayne Anderson
Copyright © 2012 Pearson Education Inc.
Goals for Chapter 15
• To study the properties and varieties of mechanical waves
• To relate the speed, frequency, and wavelength of periodic waves
• To interpret periodic waves mathematically
• To calculate the speed of a wave on a string
• To calculate the energy of mechanical waves
• To understand the interference of mechanical waves
• To analyze standing waves on a string
• To investigate the sound produced by stringed instruments
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Types of mechanical waves
• A mechanical wave is a disturbance traveling through a medium.
• Figure 15.1 below illustrates transverse waves and longitudinal
waves.
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Periodic waves
• For a periodic wave, each particle of the medium
undergoes periodic motion. The speed of the
wave is not the same as the speed of the particles.
• The wavelength  of a periodic wave is the length
of one complete wave pattern.
• The speed of any periodic wave of frequency f is
v = f.
• Example 15.1: The speed of sound in air at 20° C is 344
m/s. What is the wavelength of a sound wave in air at 20° C if
the frequency is 262 Hz?
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Periodic transverse waves
•
For the transverse waves shown here in
Figures 15.3 and 15.4, the particles move
up and down, but the wave moves to the
right.
•
This difference in direction of the waves
and particles is why the wave is called a
transverse wave.
•
Note that the restoring force is transverse
to the direction of the wave propagation.
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Periodic longitudinal waves
• For the longitudinal waves shown
here in Figures 15.6 and 15.7, the
particles oscillate back and forth
along the same direction that the
wave moves.
• The restoring force (pressure) is
in the same direction as the wave
propagation.
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Mathematical description of a wave
• The wave function, y(x,t), gives a
mathematical description of a wave. In
this function, y is the displacement of a
particle at time t and position x.
• The wave function for a sinusoidal
wave moving in the +x-direction is
y(x,t) = Acos(kx – t), where k = 2π/
is called the wave number.
• For transverse waves, y might
represent the height of the wave at
location x.
• For longitudinal waves, y might
represent the pressure at location x.
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Graphing the wave function
• The graphs in Figure 15.9 to the right look
similar, but they are not identical. Graph
(a) shows the shape of the string at t = 0,
but graph (b) shows the displacement y as
a function of time at t = 0.
• Starting with y(x = 0, t) = A cost, the
wave moving at speed v to the right will
cause the motion at a point x to be delayed
by time t = x/v. Thus
  x 
  x 
y ( x, t )  A cos   t     A cos    t  
  v 
  v 
• But  = v/f = 2 v/, and T = 2 / so
 x

y ( x, t )  A cos  2    t 
 

• We define k = 2/ = wavenumber, so
y( x, t )  A cos  kx  t 
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wave function
Derivatives of y: wave equation
• Starting with y( x, t )  A cos  kx  t  , take partial derivative with respect to
time to get y component of velocity:
v y ( x, t ) 
y ( x, t )
  A sin  kx  t 
t
• Likewise, take another partial derivative to get y component of acceleration:
 2 y ( x, t )
2
2
a y ( x, t ) 



A
cos
kx


t



y ( x, t )


2
t
• We can also take partial derivatives with respect to x (instead of t) to get:
 2 y ( x, t )
2
2


k
A
cos
kx


t


k
y ( x, t )


2
x
• If we take the ratio of these two equations, we have:
 2 y( x, t ) / t 2  2
2
 2 y( x, t ) / x 2

k2
v
• Rearranging gives the wave equation:
 2 y ( x, t ) 1  2 y ( x , t )
 2
x 2
v
t 2
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Particle velocity and acceleration in a sinusoidal wave
• The graphs in Figure 15.10 below show the velocity and
acceleration of particles of a string carrying a transverse
wave.
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The speed of a wave on a string
•
When a portion of a string under tension is displaced, a
wave will be launched along the string.
•
The text gives two explanations of how to derive the
velocity of the wave, but I find both to be relatively
tedious and uninspiring.
•
It is probably not surprising that the velocity will depend
on the tension force F and on the weight of the string
(actually the mass/unit length m). Think of a guitar whose
strings are all at the same tension. The thicker strings will
produce a lower tone due to the slower speed of the wave
(we will discuss standing waves shortly).
•
F
Here is the expression for the propagation speed: v  m .
•
Example 15.3: One end of a 2.00 kg rope is tied to a
support at the top of a mine shaft 80.0 m deep. The rope is
stretched taut by a 20.0 kg box of rocks at the bottom.
a)
What is the speed of a transverse wave on the rope?
b)
If a point on the rope is in transverse SHM with f = 2.00
Hz, how many cycles of the wave are there in the rope’s
length?
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Boundary conditions
• When a wave reflects
from a fixed end, the
pulse inverts as it
reflects. See Figure
15.19(a) at the right.
• When a wave reflects
from a free end, the
pulse reflects without
inverting. See Figure
15.19(b) at the right.
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Wave interference and superposition
• Interference is the
result of overlapping
waves.
• Principle of superposition: When two
or more waves
overlap, the total
displacement is the
sum of the displacements of the
individual waves.
• Study Figures 15.20
and 15.21 at the
right.
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Standing waves on a string
• Waves traveling in opposite directions on a taut string
interfere with each other.
• The result is a standing wave pattern that does not move
on the string.
• Destructive interference occurs where the wave
displacements cancel, and constructive interference
occurs where the displacements add.
• At the nodes no motion occurs, and at the antinodes the
amplitude of the motion is greatest.
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Photos of standing waves on a string
• Some time exposures of standing waves on a stretched string.
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The formation of a standing wave
• In Figure 15.24, a wave to the left
combines with a wave to the right
to form a standing wave.
• Both are moving waves, one in the
+x direction and one in the –x
direction, but their sum creates a
standing wave.
y( x, t )  y1 ( x, t )  y2 ( x, t )
 A[ cos(kx  t )  cos  kx  t ]
• Using the identities for cosine of
the sum and difference of two
angles
cos(a  b)  cos a cos b sin a sin b
we easily find the standing wave
equation
y ( x, t )  (2 A sin kx)sin t
• Why is this not a traveling wave?
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Normal modes of a string
• For a taut string fixed at
both ends, the possible
wavelengths are n = 2L/n
and the possible frequencies
are fn = n v/2L = nf1, where
n = 1, 2, 3, …
• f1 is the fundamental
frequency, f2 is the second
harmonic (first overtone), f3
is the third harmonic
(second overtone), etc.
• Figure 15.26 illustrates the
first four harmonics.
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Standing waves and musical instruments
• A stringed instrument is tuned to the correct frequency (pitch) by
varying the tension. Longer strings produce bass notes and
shorter strings produce treble notes.
• For a stringed instrument with f1 = v/2L, since v  F
m , the
fundamental frequency (pitch) of a string is
f1 
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1
2L
F
m