Analytical Physics 1B Lecture 5:
Physical Pendulums and Introduction to
Mechanical Waves
John Paul Chou
Friday, February 17th, 2017
SIMPLE PENDULUMS
• Consider the torque about the point
where the string is hanging
⌧ = mgL sin ✓ = I↵
• when θ is small sin(θ)≈θ
• what is I? Remember that I is defined
about the axis of rotation
I = mL2
• Therefore,
2
mgL✓ = mL ↵
• Notice that α is always in the opposite
direction as θ, so
2
2
↵=
d ✓
dt2
Physics 124 – Physical Pendulums and Mechanical Waves
d ✓
=
2
dt
g
✓
L
This is SHM!
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SIMPLE PENDULUMS
• Given the equation of motion, let’s
derive θ as a function of time
2
d ✓
=
2
dt
g
✓
L
• Our guess is:
✓ = A sin(!t + )
• Plugging this in to the above yields:
2
A! sin(!t + ) =
• which means…
g
sin(!t + )
L
!=
Physics 124 – Physical Pendulums and Mechanical Waves
r
g
L
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PHYSICAL PENDULUMS
• Physical pendulums are like
simple pendulums but where the
mass is extended in space
• similar calculation as before:
⌧ = mgd sin ✓ = I↵
• approximate sin(θ)≈θ
• but can’t simplify expression for I!
mgd
✓=
I
!=
r
2
d ✓
dt2
mgd
I
Physics 124 – Physical Pendulums and Mechanical Waves
Don’t forget: I is with respect to the pivot point!
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DAMPED SPRINGS
• Suppose we have a spring, but there is a damping force
proportional to its velocity:
F =
kx
d2 x
m 2 =
dt
bv
kx
dx
b
dt
• the solution to this differential equation (when b is small) is…
x = Ae
(b/2m)t
• and
0
! =
r
k
m
0
cos(! t + )
b2
4m2
• What does this look like?
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DAMPED SPRINGS
x = Ae
!0 =
(b/2m)t
r
k
m
0
cos(! t + )
b2
4m2
• Notice that the amplitude is
not a constant (it is
exponentially decreasing)
• Also the frequency is
somewhat smaller
• What happens when b is
big? Overdamping.
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FORCED OSCILLATIONS
• If we apply a periodically varying force to an oscillator where
the frequency of the varying force is close to the natural
frequency of the oscillator (√k/m), then we can get
resonance behavior
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FORCED OSCILLATIONS VIRTUAL DEMO
• https://phet.colorado.edu/sims/resonance/
resonance_en.html
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ONE MORE EXAMPLE OF SHM
• Consider an object freely falling through the center of the
Earth
• It turns out that the object at radius r feels the force of gravity only
from the part of the Earth within a spherical region of radius r
• Moreover, it feels it as if it were a point mass concentrated at the
center of the earth
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ONE MORE EXAMPLE OF SHM
• Let’s assume that the Earth has a uniform
density
M
⇢=
E
3
4/3⇡RE
• Therefore the mass contained within a sphere of
radius r is
3
r
M (r) = ⇢4/3⇡r = ME 3
RE
3
• So the force felt by an object at a given radius
must be
GmM (r)
GmME
F =
=
r
3
2
r
RE
Physics 124 – Physical Pendulums and Mechanical Waves
a=
GME
r
3
RE
This is SHM!
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ONE MORE EXAMPLE OF SHM
• We can calculate the period of the motion:
!=
s
GME
3
RE
T = 2⇡
s
3
RE
GME
• Plugging in the constants, we get that the period is 84 minutes
• Therefore, it would take a period 42 minutes to freely fall from one
side of the earth to the other, just using the force of gravity
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MECHANICAL WAVES
• Waves resulting from the physical displacement of part of
the medium from equilibrium
• These are to be distinguished from say electromagnetic waves
or gravitational waves
• These wave can be transverse or longitudinal
• transverse waves – particles
undergo displacements in a
direction perpendicular to the
direction of wave motion
• longitudinal waves – particles
undergo displacements in a
direction parallel to the direction
of wave motion
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PERIODIC TRANSVERSE WAVES
• Any periodic wave can be expressed as the sum of many
sinusoidal waves
• The waves have an amplitude, a frequency, a wavelength, and a
speed with which they propagate
• the wave advances one wavelength in a time interval of one period
!
= f=
v=
T
2⇡
wavelength
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PERIOD TRANSVERSE WAVES
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RANKING FREQUENCIES
The drawings represent snapshots taken of waves traveling to the right along strings.
The grids shown in the background are identical. The waves all have the same speed,
but their amplitudes and wavelengths vary. Rank the frequency of the waves.
A.
B.
C.
D.
E.
B>A>D>C
B>A>D=C
They are all the same.
C>D>A>B
C=D>A>B
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RANKING FREQUENCIES
The drawings represent snapshots taken of waves traveling to the right along strings.
The grids shown in the background are identical. The waves all have the same speed,
but their amplitudes and wavelengths vary. Rank the frequency of the waves.
A.
B.
C.
D.
E.
B>A>D>C
B>A>D=C
They are all the same.
C>D>A>B
C=D>A>B
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RANKING SPEEDS
The drawings represent snapshots taken of waves traveling to the right along strings. The
grids shown in the background are identical. The waves all have the same frequency, but
their amplitudes and wavelengths vary. Rank the speed of the waves on the string.
A.
B.
C.
D.
E.
B>A>D>C
B>A>D=C
They are all the same.
C>D>A>B
C=D>A>B
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RANKING SPEEDS
The drawings represent snapshots taken of waves traveling to the right along strings. The
grids shown in the background are identical. The waves all have the same frequency, but
their amplitudes and wavelengths vary. Rank the speed of the waves on the string.
A.
B.
C.
D.
E.
B>A>D>C
B>A>D=C
They are all the same.
C>D>A>B
C=D>A>B
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MATHEMATICAL DESCRIPTION OF A WAVE
• Consider a wave propagating on a
string extended in the x direction
• The position in y of any point on the string
depends on its location in x and what time
it is
• i.e. the wave function depends on x and t:
y(x,t)
• Consider the particle at point B
• it starts at the maximum value of y at t=0,
and returns to it after t=T:
y(x = xB , t) = A cos !t
• where
! = 2⇡/T
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MATHEMATICAL DESCRIPTION OF A WAVE
• We solved for a single point x=B, but
how does y depend in general on x?
• Notice that this wave is moving left to
right
• As we know, the wave travels with a
speed λf, which means that:
• the motion of point B at time t is the
same as the motion of point A at the
earlier time t-x/v, where x is the distance
between A and B
y(x = 0, t) = A cos !t
h ⇣
y(x, t) = A cos ! t
Physics 124 – Physical Pendulums and Mechanical Waves
x ⌘i
v
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MATHEMATICAL DESCRIPTION OF A WAVE
• We can simplify the expression so that
y(x, t) = A cos(kx
!t)
• where k a quantity we define as the wave
number
k⌘
2⇡
• For a wave traveling right to left:
y(x, t) = A cos(kx + !t)
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DEFINITIONS
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PRINCIPLE OF SUPERPOSITION
• When two waves overlap, the actual
displacement on the string is given
by adding the together the expect
locations of the waves had the other
wave not been present
y(x, t) = y1 (x, t) + y2 (x, t)
• Waves can interact constructively or
destructively, resulting in a potentially
larger or smaller final wave than either
of the two input waves
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COLLIDING WAVES
Rectangular transverse wave pulses are traveling toward each other along a string. The
grids shown in the background are identical, and the pulses vary in height and length.
The pulses will meet and interact soon after they are in the positions shown. Rank the
maximum amplitude of the string at the instant that the positions of the centers of
the two pulses coincide.
A.
B.
C.
D.
E.
C>D>A>B
A>C>B>D
They are all the same.
A>B>C>D
A=C>B=D
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COLLIDING WAVES
Rectangular transverse wave pulses are traveling toward each other along a string. The
grids shown in the background are identical, and the pulses vary in height and length.
The pulses will meet and interact soon after they are in the positions shown. Rank the
maximum amplitude of the string at the instant that the positions of the centers of
the two pulses coincide.
A.
B.
C.
D.
E.
C>D>A>B
A>C>B>D
They are all the same.
A>B>C>D
A=C>B=D
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LIGO
• https://www.youtube.com/watch?v=s06_jRK939I
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