pendulum lecturer: professor stephen t. thornton
TRANSCRIPT
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Pendulum
Lecturer: Professor Stephen T. Thornton
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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.
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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.
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Last Time
Oscillations
Simple harmonic motion
Periodic motion
Springs
Energy
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Today
Simple pendulum
Physical pendulum
Damped and forced oscillations
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Motion of a Pendulum
0U
(1 cos )U mgL
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Small Angles
24
22
(1 cos )
Small angles: cos 1 ( )2
1 (1 )2 2
U mgL
O
mgLU mgL
This is a parabola. Pendulum has similar potential energy to a spring.
0
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The Potential Energy of a Simple Pendulum
(1 cos )U mgL
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The Simple Pendulum
Position of mass along arc:
Velocity along the arc:
Tangential acceleration:
s
ds dvdt dt
22
dv dadt dt
L
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The tangential restoring force comes from gravity (tension is always centripetal for a pendulum):
tan
tan sin
But = , and / , so we have
= mg
F mg mg
x xmgF x k
We have a restoring force F = -kx for small angle oscillations, which is like Hooke’s law, so we have simple harmonic motion!
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max
1
Let's find the solution in terms of , not .
cos( )
where
so we have
12 21 2
Remember this is all true for small angles .
x
t tmg gk
m
gf
T gf
mg
//
xk mg
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The Simple Pendulum
2Tg
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:
h2 2
2 22
22 2
( ) (1 cos )
1 1( ) 1 (1 ...)2 2
1 1 1( )2 2 2
1 1( ) and ( )2 2
U mgh mg
U mg mg
dx dK mv m mdt dt
dK m U mgdt
x
http://physics.bu.edu/~duffy/semester1/semester1.html
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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.
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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.
2Tg
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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) upB) downC) moving the weight does not matter.D) throw the clock away and get a new one, because physics is too hard.
2Tg
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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.
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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?
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Examples of Physical Pendulums
Demo
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A physical pendulum is any real extended object that oscillates back and forth.
The torque about point O is:
Substituting into Newton’s second law for rotation gives:
The Physical Pendulum
sinmght q=-
2
2sin
dI mgh
dt
qq=-
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For small angles, this becomes:
which is the equation for SHM, with
max cos( )
2
mghI
t
ITmgh
2
20
d mgh
dt I
æ ö÷ç+ =÷ç ÷çè ø
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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) both the period and the energy are unchanged.B) both the period and the energy increase.C) the period is unchanged and the energy increases.D) the period increases and the energy is unchanged.E) none of these is correct.
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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.
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Conceptual QuizConceptual Quiz
A hole is drilled through the
center of Earth and emerges
on the other side. You jump
into the hole. What happens
to you ?
A) you fall to the center and stop
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
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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 gravity pulls
you back toward the centeryou back toward the center. This is Simple This is Simple
Harmonic Motion!Harmonic Motion!
Conceptual QuizConceptual Quiz
A hole is drilled through the
center of Earth and emerges
on the other side. You jump
into the hole. What happens
to you ?
A) you fall to the center and stop
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
Follow-up:Follow-up: Where is your acceleration zero? Where is your acceleration zero?
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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 QuizConceptual Quiz
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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
elastic potential energy, which is PEs = kA2. This
does not depend on mass, so a change in mass will not affect the energy of the system.
Conceptual QuizConceptual Quiz
Follow-up:Follow-up: What happens if you double the amplitude? What happens if you double the amplitude?
12
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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.
Damped Harmonic Motion
dampingIf ,
then is Newton's 2nd law
F bv
ma kx bv
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This gives
If b is small, a solution of the form
will work, with
Damped Harmonic Motion
2
20
d x dxm b kx
dt dt+ + =
cos 'tx Ae tg w-=
2
2
2
'4
b
m
k b
m m
g
w
=
= -
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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 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.
Forced Oscillations; Resonance
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The equation of motion for a forced oscillator is:
The solution is:
where
and
0 cosma kx bv F tw=- - +
0 0sin( )x A tw f= +
( )0
0 22 2 2 2 20 /
FA
m b mw w w=
- +
( )
2 21 0
0 tan/b m
w wf
w-æ ö- ÷ç ÷= ç ÷ç ÷÷çè ø
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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.
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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.
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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.
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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?