Transcript simple harmonic motion

Vibrations and Waves
Topics:
 Equilibrium, restoring
forces, and oscillation
 Energy in oscillatory
motion
 Mathematical description
of 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?
Where in nature do we encounter vibrations?
How are vibrations related to waves?
What does it mean for motion to be periodic?
When a vibration repeats itself, the motion is periodic.
The simplest periodic motion is spring motion.
Many systems resemble spring motion.
Oscillation requires an equilibrium and a restoring force
The object’s displacement
varies with time.
Its maximum displacement
is called its amplitude.
Simple Harmonic Motion
Periodic Motion: When an object repeats the same motion over and
over, we refer to the motion as periodic motion or oscillatory motion.
A mechanical system that undergoes periodic motion always has a stable
equilibrium position.
The system will also always have a restoring force that acts to restore
the system to its equilibrium position.
Periodic motion under the action of a restoring force that is directly
proportional to the displacement from equilibrium is called
simple harmonic motion (SHM).
By “directly proportional” we mean linear. In symbols, we have:
Frestore x ∝ Δx
Simple Harmonic Motion
But the restoring force won’t do much restoring unless it opposes the
motion away from equilibrium.
So Frestore x and Δx need to have opposite signs.
Putting this together we have the condition for SHM:
Frestore x = Fx = – kΔx = – kx.
This is Hooke’s Law.
The constant k is called the spring constant. It is a measure of the
stiffness of the spring. The SI units for k are N/m.
In other SHM contexts, this can be generalized, and k will be replaced by
an analogous quantity.
Simple Harmonic Motion
simple harmonic motion (SHM)
farther mass “runs away”
↔
↔
linear restoring force
more restoring force
if mass goes right ↔ force points left
if mass goes left ↔ force points right
Energy and Simple Harmonic Motion
Energy and Simple Harmonic Motion
Energy and Simple Harmonic Motion
Energy and Simple Harmonic Motion
Energy and Simple Harmonic Motion
SHM Period and Frequency
The frequency of oscillation depends on physical properties of the oscillator;
it does not depend on the amplitude of the oscillation.
Position, Velocity, and Acceleration
Are Sinusoidal Functions of Time
Damping
Damping
Resonance
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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Sinusoidal Relationships
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Mathematical Description of Simple Harmonic Motion
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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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Solving Problems
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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.
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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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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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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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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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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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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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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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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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Damping
A 500 g mass on a string oscillates as a pendulum. The
pendulum’s energy decays to 50% of its initial value in 30 s.
What is the value of the damping constant?
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Resonance
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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.
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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.
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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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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?
B. 3.3 s
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Additional Example
Walter has a summer job babysitting a 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 Examples
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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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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Answer
1. The type of function that describes simple harmonic motion is
D. sinusoidal
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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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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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Reading Quiz
3. If you drive an oscillator, it will have the largest amplitude if you
drive it at its _______ frequency.
A. special
B. positive
C. resonant
D. damped
E. pendulum
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Answer
3. If you drive an oscillator, it will have the largest amplitude if you
drive it at its _______ frequency.
C. resonant
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