Lecture 31.Pendulum

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Pendulum
Lecturer:
Professor Stephen T. Thornton
Reading Quiz
What is happening to the
bridge in this photo?
A) A ship passing under the bridge has just hit it.
B) This is a fake photo for one of the thriller
movies.
C) The wind is causing a forced resonant
oscillation.
D) This is a painting, not a photo.
C)
This is a photo of the Tacoma
Narrows bridge collapse of the 1940s.
We will watch a video of it today.
The bridge oscillated in resonance
and eventually broke apart.
Last Time
Oscillations
Simple harmonic motion
Periodic motion
Springs
Energy
Today
Simple pendulum
Physical pendulum
Damped and forced oscillations
Motion of a Pendulum
U  m gL (1  cos  )
U 0
Small Angles
U  m gL (1  cos  )
S m all angles: cos   1 

0
2
 O ( )
4
2
2

  m gL
U  m gL 1  (1 
) 

2 
2

2
This is a parabola. Pendulum has
similar potential energy to a spring.
The Potential Energy of a Simple Pendulum
U  m gL (1  cos  )
The Simple Pendulum
Position of mass along arc:
s 
Velocity along the arc:
v  ds 
dt
d
dt
Tangential acceleration:
2
dv
d
a

dt
dt 2
L
The tangential restoring force
comes from gravity (tension is
always centripetal for a
pendulum):
F tan   m g sin    m g 
B ut x =  , and   x / , so w e have
F tan =  m g x
k  mg
We have a restoring force F = -kx for small angle
oscillations, which is like Hooke’s law, so we
have simple harmonic motion!
L et's find the solution in term s of  , no t x.
   m ax cos( t  t )
k  mg 1  g
w here   m
m
so w e have
  x/
k  mg /
g

g

1
f 

2 2
T  1  2 g
f
R em em ber this is all true for sm all angles  .
The Simple Pendulum
T  2
g
This is a remarkable result.
The period only depends
on the length of the
pendulum, not the mass!
Galileo figured this out as a
young man sitting in
church while watching the
chandeliers swing.
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Energy of a simple pendulum:
U ( )  m gh  m g (1  cos )












U ( )  m g 1 (1 1  2  ...)  1 m g  2
2
2






K ( )  1 m v 2  1 m dx
2
2
dt






K ( )  1 m 2 d 
2
dt












2






 1 m d
2
dt






2
2
and U ( )  1 m g  2
2
http://physics.bu.edu/~duffy/semester1/semester1.html
h
x 
Conceptual Quiz:
A person sits on a playground swing.
When pushed gently once, the swing
oscillates back and forth at its natural
frequency. If, instead, two people sit side
by side on the swing, the new natural
frequency of the swing is
A) greater.
B) smaller.
C) the same.
Answer: C
The problem statement indicated it is
a gentle push, so we assume small
oscillations. In that case, the period
doesn’t depend on the mass, only the
length of the swing.
T  2
g
Conceptual Quiz:
Grandfather clocks have a weight at the
bottom of the pendulum arm that can be
moved up or down to correct the time.
Suppose that your grandfather clock runs
slow. In which direction do you move the
weight to correct the time on the clock?
A)
B)
C)
D)
T  2
up
g
down
moving the weight does not matter.
throw the clock away and get a new one,
because physics is too hard.
A) up
In order for the clock to run faster, we
want the time between ticks to be smaller.
That is, we want the period to decrease. In
order to do that we decrease L which
decreases the period. We adjust a small
screw usually on the bottom of the pendulum
arm that raises the weight (mass bob). This
decreases L and makes the clock run faster.
Simple Pendulum. What is the
period of a simple pendulum 53
cm long (a) on the Earth, and (b)
when it is in a freely falling
elevator?
Examples of Physical Pendulums
D em o
The Physical Pendulum
A physical pendulum is any real
extended object that oscillates
back and forth.
The torque about point O is:
t = - m gh sin q
Substituting into Newton’s second
law for rotation gives:
2
I
d q
dt
2
= - m gh sin q
For small angles, this becomes:
2
d q
dt
2
æm g h ö
÷
+ çç
q= 0
÷
÷
çè I ø
which is the equation for SHM, with
m gh

I
   m ax cos( t   )
I
T  2
m gh
Conceptual Quiz:
A simple pendulum oscillates with a maximum
angle to the vertical of 5o. If the same pendulum is
repositioned so that its maximum angle is 7o, we can
say that
A)
B)
C)
D)
E)
both the period and the energy are unchanged.
both the period and the energy increase.
the period is unchanged and the energy increases.
the period increases and the energy is unchanged.
none of these is correct.
Answer: C
This is a small oscillation, and for
small oscillations, the period does not
change significantly.
The weight moves further up in
elevation, and its U increases, so its
total energy also increases.
Conceptual Quiz
A hole is drilled through the
A) you fall to the center and stop
center of Earth and emerges
on the other side. You jump
into the hole. What happens
to you ?
B) you go all the way through and
continue off into space
C) you fall to the other side of
Earth and then return
D) you won’t fall at all
Conceptual Quiz
A hole is drilled through the
A) you fall to the center and stop
center of Earth and emerges
on the other side. You jump
into the hole. What happens
to you ?
B) you go all the way through and
continue off into space
C) you fall to the other side of
Earth and then return
D) you won’t fall at all
You fall through the hole. When you reach the
center, you keep going because of your inertia.
When you reach the other side, gravity pulls
you back toward the center. This is Simple
Harmonic Motion!
Follow-up: Where is your acceleration zero?
Conceptual Quiz
A mass oscillates in simple
harmonic motion with amplitude
A. If the mass is doubled, but the
amplitude is not changed, what
will happen to the total energy of
the system?
A) total energy will increase
B) total energy will not change
C) total energy will decrease
Conceptual Quiz
A mass oscillates in simple
harmonic motion with amplitude
A. If the mass is doubled, but the
amplitude is not changed, what
will happen to the total energy of
the system?
A) total energy will increase
B) total energy will not change
V) total energy will decrease
The total energy is equal to the initial value of the
1
elastic potential energy, which is PEs = 2 kA2. This
does not depend on mass, so a change in mass will
not affect the energy of the system.
Follow-up: What happens if you double the amplitude?
Damped Harmonic Motion
Damped harmonic motion is harmonic
motion with a frictional or drag force. If the
damping is small, we can treat it as an
“envelope” that modifies the undamped
oscillation.
If Fdam ping   bv ,
then
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m a   kx  bv
is N ew ton's 2nd law
Damped Harmonic Motion
This gives
2
m
d x
dt
2
+ b
dx
+ kx = 0
dt
If b is small, a solution of the form
x = Ae
will work, with
g =
- gt
cos w ' t
b
2m
w'=
k
m
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-
b
2
4m
2
If b2 > 4mk, ω’ becomes imaginary, and
the system is overdamped (C).
For b2 = 4mk, the system is critically damped (B)
—this is the case in which the system reaches
equilibrium in the shortest time.
Case A (b2 < 4mk)
is underdamped; it
oscillates within the
exponential
envelope.
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There are systems in which
damping is unwanted,
such as clocks and
watches.
Then there are systems in
which it is wanted, and
often needs to be as close
to critical damping as
possible, such as
automobile shock
absorbers, storm door
closures, and earthquake
protection for buildings..
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Forced Oscillations; Resonance
Forced vibrations occur when there is a periodic
driving force. This force may or may not have the
same period as the natural frequency of the
system.
If the frequency is the same as the natural frequency,
the amplitude can become quite large. This is called
resonance.
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The equation of motion for a forced oscillator
is:
m a = - kx - bv + F0 cos w t
The solution is:
x = A0 sin( w t + f 0 )
F0
A0 =
where
m
and
f
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0
(w
2
- w
2
0
2
)
2
2
+ b w /m
æw 2 - w 2 ö
÷
- 1ç
0
÷
= tan çç
÷
çè w (b / m )÷
ø
2
Show hacksaw blade resonance
demo. (Go back and show previous slide.)
Do damping and forced oscillation
demo. (Go back and show previous slide.)
Show Tacoma Narrows Bridge
collapse.
The sharpness of the
resonant peak depends
on the damping. If the
damping is small (A) it
can be quite sharp; if the
damping is larger (B) it
is less sharp.
Like damping, resonance can be wanted or
unwanted. Musical instruments and TV/radio
receivers depend on it.
Copyright © 2009 Pearson Education, Inc.
Human Leg. The human leg can be compared to
a physical pendulum, with a “natural” swinging
period at which walking is easiest. Consider the
leg as two rods joined rigidly together at the knee;
the axis for the leg is the hip joint. The length of
each rod is about the same, 55 cm. The upper rod
has a mass of 7.0 kg and the lower rod has a mass
of 4.0 kg. (a) Calculate the natural swinging
period of the system. (b) Check your answer by
standing on a chair and measuring the time for
one or more complete back-and-forth swings. The
effect of a shorter leg is a shorter swinging period,
enabling a faster “natural” stride.
Unbalanced Tires. An 1150 kg
automobile has springs with k =
16,000 N/m. One of the tires is
not properly balanced; it has a
little extra mass on one side
compared to the other, causing the
car to shake at certain speeds. If
the tire radius is 42 cm, at what
speed will the wheel shake most?
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