Transcript 14_LectureOutlineOsc..

Chapter 14
Oscillations
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PowerPoint® Lectures for
College Physics: A Strategic Approach, Second Edition
14 Oscillations
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Slide 14-2
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Slide 14-3
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Equilibrium and Oscillation
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Slide 14-12
Linear Restoring Forces and Simple Harmonic
Motion
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Frequency and Period
The frequency of oscillation depends on physical properties of the
oscillator; it does not depend on the amplitude of the oscillation.
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Checking Understanding
A set of springs all have initial length 10 cm. Each spring now has
a mass suspended from its end, and the different springs stretch
as shown below.
Now, each mass is pulled down by an additional 1 cm and
released, so that it oscillates up and down. Rank the frequencies
of the oscillating systems A, B, C and D, from highest to lowest.
A.
B.
C.
D.
BDCA
BADC
CADB
ACBD
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Answer
A set of springs all have initial length 10 cm. Each spring now has
a mass suspended from its end, and the different springs stretch
as shown below.
Now, each mass is pulled down by an additional 1 cm and
released, so that it oscillates up and down. Rank the frequencies
of the oscillating systems A, B, C and D, from highest to lowest.
A.
B.
C.
D.
BDCA
BADC
CADB
ACBD
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Slide 14-16
Checking Understanding
A series of pendulums with different length strings and different
masses is shown below. Each pendulum is pulled to the side by
the same (small) angle, the pendulums are released, and they
begin to swing from side to side.
Rank the frequencies of the five pendulums, from highest to
lowest.
A.
B.
C.
D.
AEBDC
DACBE
ABCDE
BECAD
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Answer
A series of pendulums with different length strings and different
masses is shown below. Each pendulum is pulled to the side by
the same (small) angle, the pendulums are released, and they
begin to swing from side to side.
Rank the frequencies of the five pendulums, from highest to
lowest.
A.
B.
C.
D.
AEBDC
DACBE
ABCDE
BECAD
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Example Problems
The first astronauts to visit Mars are each allowed to take along
some personal items to remind them of home. One astronaut
takes along a grandfather clock, which, on earth, has a pendulum
that takes 1 second per swing, each swing corresponding to one
tick of the clock. When the clock is set up on Mars, will it run fast
or slow?
A 5.0 kg mass is suspended from a spring. Pulling the mass down
by an additional 10 cm takes a force of 20 N. If the mass is then
released, it will rise up and then come back down. How long will it
take for the mass to return to its starting point 10 cm below its
equilibrium position?
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Energy in Simple Harmonic Motion
As a mass on a spring goes through its cycle of oscillation, energy
is transformed from potential to kinetic and back to potential.
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Sinusoidal Relationships
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Mathematical Description of Simple Harmonic
Motion
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Example Problem
A ball on a spring is pulled down and then released. Its
subsequent motion appears as follows:
1.
2.
3.
4.
5.
6.
At which of the above times is the displacement zero?
At which of the above times is the velocity zero?
At which of the above times is the acceleration zero?
At which of the above times is the kinetic energy a maximum?
At which of the above times is the potential energy a maximum?
At which of the above times is kinetic energy being transformed to
potential energy?
7. At which of the above times is potential energy being transformed
to kinetic energy?
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Example Problem
A pendulum is pulled to the side and released. Its subsequent
motion appears as follows:
1.
2.
3.
4.
5.
6.
At which of the above times is the displacement zero?
At which of the above times is the velocity zero?
At which of the above times is the acceleration zero?
At which of the above times is the kinetic energy a maximum?
At which of the above times is the potential energy a maximum?
At which of the above times is kinetic energy being transformed to
potential energy?
7. At which of the above times is potential energy being transformed
to kinetic energy?
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Solving Problems
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Example Problem
We think of butterflies and moths as gently fluttering their wings,
but this is not always the case. Tomato hornworms turn into
remarkable moths called hawkmoths whose flight resembles that
of a hummingbird. To a good approximation, the wings move with
simple harmonic motion with a very high frequency—about 26 Hz,
a high enough frequency to generate an audible tone. The tips of
the wings move up and down by about 5.0 cm from their central
position during one cycle. Given these numbers,
A.
What is the maximum velocity of the tip of a hawkmoth
wing?
B.
What is the maximum acceleration of the tip of a
hawkmoth wing?
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Example Problem
A car rides on four wheels that are connected to the body of the
car by springs that allow the car to move up and down as the
wheels go over bumps and dips in the road. Each spring supports
approximately 1/4 the mass of the vehicle. A lightweight car has a
mass of 2400 lbs. When a 160-lb person sits on the left front
fender, this corner of the car dips by about 1/2 in.
A.
B.
What is the spring constant of this spring?
When four people of this mass are in the car, what is the
oscillation frequency of the vehicle on the springs?
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Example Problem
Manufacturers are now making shoes with springs in the soles. A
spring in the heel will decrease the impact force when your heel
strikes, but, equally important, it can store energy that is returned
when the foot rolls forward to push off. Ideally, a shoe would be
designed so that, when your heel strikes, the spring compresses
and then rebounds at a natural point in your stride. We can model
this rebound as a segment of an oscillation to get some idea of
how such a shoe should be designed.
A.
In a moderate running gait, your heel is in contact with
the ground for 0.1 s or so. For optimal timing, what
should the period of the motion for the mass (the runner)
and spring (the spring in the shoe) system be?
B. What should be the spring constant of the spring for a
70-kg man?
C. When the man puts all his weight on the heel, how much
should the spring compress?
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Example Problem
A 204 g block is suspended from a vertical spring, causing the
spring to stretch by 20 cm. The block is then pulled down an
additional 10 cm and released. What is the speed of the block
when it is 5.0 cm above the equilibrium position?
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Damping
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Resonance
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Summary
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Summary
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Additional Questions
Four different masses are hung from four springs with unstretched
length 10 cm, causing the springs to stretch as noted in the
following diagram:
Now, each of the masses is lifted a small distance, released, and
allowed to oscillate. Rank the oscillation frequencies, from highest
to lowest.
A. a  b  c  d
B. d  c  b  a
C. a  b  c  d
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Answer
Four different masses are hung from four springs with unstretched
length 10 cm, causing the springs to stretch as noted in the
following diagram:
Now, each of the masses is lifted a small distance, released, and
allowed to oscillate. Rank the oscillation frequencies, from highest
to lowest.
A. a  b  c  d
B. d  c  b  a
C. a  b  c  d
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Additional Questions
Four 100 g masses are hung from four springs, each with
unstretched length 10 cm. The four springs stretch as noted in the
following diagram:
Now, each of the masses is lifted a small distance, released,
and allowed to oscillate. Rank the oscillation frequencies, from
highest to lowest.
A. a  b  c  d
B. d  c  b  a
C. a  b  c  d
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Answer
Four 100 g masses are hung from four springs, each with
unstretched length 10 cm. The four springs stretch as noted in the
following diagram:
Now, each of the masses is lifted a small distance, released,
and allowed to oscillate. Rank the oscillation frequencies, from
highest to lowest.
A. a  b  c  d
B. d  c  b  a
C. a  b  c  d
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Additional Questions
A pendulum is pulled to the side and released. Rank the following
positions in terms of the speed, from highest to lowest. There may
be ties.
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Additional Questions
A typical earthquake produces vertical oscillations of the earth.
Suppose a particular quake oscillates the ground at a frequency
0.15 Hz. As the earth moves up and down, what time elapses
between the highest point of the motion and the lowest point?
A. 1 s
B. 3.3 s
C. 6.7 s
D. 13 s
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Answer
A typical earthquake produces vertical oscillations of the earth.
Suppose a particular quake oscillates the ground at a frequency
0.15 Hz. As the earth moves up and down, what time elapses
between the highest point of the motion and the lowest point?
A. 1 s
B. 3.3 s
C. 6.7 s
D. 13 s
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Additional Example Problem
Walter has a summer job babysitting an 18 kg youngster. He takes
his young charge to the playground, where the boy immediately
runs to the swings. The seat of the swing the boy chooses hangs
down 2.5 m below the top bar. “Push me,” the boy shouts, and
Walter obliges. He gives the boy one small shove for each period of
the swing, in order keep him going. Walter earns $6 per hour. While
pushing, he has time for his mind to wander, so he decides to
compute how much he is paid per push. How much does Walter
earn for each push of the swing?
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Additional Example Problems
A 500 g block is attached to a spring on a frictionless horizontal
surface. The block is pulled to stretch the spring by 10 cm, then
gently released. A short time later, as the block passes through the
equilibrium position, its speed is 1.0 m/s. What is the block’s
period of oscillation? What is the block’s speed at the point where
the spring is compressed by 5.0 cm?
A mass bounces up and down on a spring. The oscillation decays
with a time constant of 50 s. If the oscillation begins with an
amplitude of 20 cm, how long will it take until the amplitude has
decreased by half to 10 cm? If the oscillation begins with an
amplitude of 20 cm, how long will it take until the energy of the
oscillation has decreased by half?
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