Transcript Powerpoint for today

Chapter 14
Oscillations
Topics:
•
Equilibrium, restoring
forces, and oscillation
•
Mathematical description of
oscillatory motion
•
•
•
Energy in oscillatory motion
Damped oscillations
Resonance
Sample question:
The gibbon will swing more rapidly and move more quickly through
the trees if it raises its feet. How can we model the gibbon’s motion
to understand this observation?
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Slide 14-1
Beyond the Elastic Limit
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Slide 8-25
Checking Understanding
The same spring is stretched or compressed as shown below.
In which case does the force exerted by the spring have the
largest magnitude?
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Slide 8-18
Answer
The same spring is stretched or compressed as shown below.
In which case does the force exerted by the spring have the
largest magnitude?
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Slide 8-19
Spring Problem 1
A 20-cm long spring is attached to the wall. When pulled
horizontally with a force of 100 N, the spring stretches to a
length of 22 cm.
What is the value of the spring constant?
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Slide 10-23
Spring Problem 2
The same spring is used in a tug-of-war. Two people pull on the
ends, each with a force of 100 N.
How long is the spring while it is being pulled?
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Slide 10-23
Spring Problem 3
The same spring is now placed vertically on the ground and a
10.2 kg block is balanced on it.
How high is the compressed spring?
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Slide 10-23
Spring Problem 4
The same spring is placed vertically on the ground and a 10.2
kg block is held 15 cm above the spring. The block is dropped,
hits the spring, and compresses it.
What is the height of the spring at the point of maximum
compression?
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Slide 10-23
Spring Problem 5
A spring with spring constant k = 125 N/m is used to pull a 25
N wooden block horizontally across a tabletop. The coefficient
of friction between the block and the table is µk = 0.20.
By how much does this spring stretch from its equilibrium
length?
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Slide 8-22
Reading Quiz
1. The type of function that describes simple harmonic motion is
A. linear
B. exponential
C. quadratic
D. sinusoidal
E. inverse
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Slide 14-2
Answer
1. The type of function that describes simple harmonic motion is
D. sinusoidal
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Slide 14-3
Reading Quiz
2. A mass is bobbing up and down on a spring. If you increase the
amplitude of the motion, how does this affect the time for one
oscillation?
A. The time increases.
B. The time decreases.
C. The time does not change.
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Slide 14-4
Answer
2. A mass is bobbing up and down on a spring. If you increase the
amplitude of the motion, how does this affect the time for one
oscillation?
C. The time does not change.
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Slide 14-5
Equilibrium and Oscillation
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Slide 14-8
Linear Restoring Forces and Simple Harmonic Motion
If the restoring force is a linear function of the displacement from
equilibrium, the oscillation is sinusoidal—simple harmonic motion.
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Slide 14-9
Sinusoidal Relationships
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Slide 14-10
Mathematical Description of Simple Harmonic Motion
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Slide 14-11
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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Slide 14-12
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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Slide 14-13
Solving Problems
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Slide 14-14
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.
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Slide 14-15
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.
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Slide 14-16
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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Slide 14-17
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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Slide 14-18
Examples
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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Slide 14-19
Example
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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Slide 14-20
Example
The deflection of the end of a diving board produces a linear
restoring force, as we saw in Chapter 8. A diving board dips by 15
cm when a 65 kg person stands on its end. Now, this person jumps
and lands on the end of the board, depressing the end by another
10 cm, after which they move up and down with the oscillations of
the end of the board.
A. Treating the person on the end of the diving board as a mass on
a spring, what is the spring constant?
B. For a 65 kg diver, what will be the oscillation period?
C. For the noted oscillation, what will be the maximum speed?
D. What amplitude would lead to an acceleration greater than that
of gravity—meaning the person would leave the board at some
point during the cycle?
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Slide 14-21
Example
In Chapter 10, we saw that the Achilles tendon will stretch and then rebound, storing and
returning energy during a step. We can model this motion as that of a mass on a spring. It’s
far from a perfect model, but it does give some insight. Suppose a 60 kg person stands on a
low wall with her full weight on the balls of one foot and the heel free to move. The stretch of
the Achilles tendon will cause her center of mass to lower by about 2.5 mm.
A. What is the value of k for this system?
B. Given the mass and the spring constant, what would you expect for the period of this
system were it to undergo an oscillation?
C. When the balls of the feet take the weight of a stride, the tendon spring begins to
stretch as the body moves down; kinetic energy is being converted into elastic potential
energy. Ideally, when the foot is leaving the ground, the cycle of the motion will have
advanced so that potential energy is being converted to kinetic energy. What fraction of
an oscillation period should the time between landing and lift off correspond to? Given
the period you calculated above, what is this time?
D. Sprinters running a short race keep their foot in contact with the ground for about 0.10
s, some of which corresponds to the heel strike and subsequent rolling forward of the
foot. Given this, does the final number you have calculated above make sense?
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Slide 14-22
Example
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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Slide 14-23