Download Simple Harmonic Motion: a system that oscillates with a constant

Oscillations
Simple Harmonic Motion: a system that oscillates with a constant frequency (or
period) is said to undergo SHM.
These terms will be used for the next few chapters
1. Cycle / Oscillation: the repeated motion of the system
2.
Period (T): time for one oscillation
3. Frequency (f or greek letter ν - “nu”): number of oscillations per second
a. T = 1/f
[s]
b. f = 1/T
[Hz] or [s-1]
4. Amplitude (A): distance from equilibrium position (no net forces on object)
can be a linear or angular distance
5. Restoring force: force that brings the oscillator back to its equilibrium position
Types of Simple Harmonic Oscillators:
A. Pendulum
1. The amplitude is the angular distance measured from the equilibrium position.
2. The restoring force is the sine component of the bob’s weight.
3. Here is a picture showing the sine component as the restoring (Fnet) force.
4. T = 2π√
𝐿
𝑔
where L is the length in [m] of the pendulum (to the bob’s center-of-mass)
and g is our old friend, 9.80-m/s2. T is in [s].
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The mass of the bob has no affect on the period, which should come as no surprise,
since mass does not affect the rate of free fall acceleration.
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The amplitude (if kept below ~ 20o) also has no affect on the period.
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Go to the middle of this
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Velocity and acceleration vectors on an oscillating
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What is a
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What is a Seconds pendulum? It is a pendulum which has a “tick” of 1.0000-s and a
“tock” of 1.0000-s, giving it a period of 2.0000-s.
page to use this variable pendulum.
pendulum.
Foucault pendulum?
B. Mass – Spring System
1. Here is a mass hooked to the bottom of a spring.
2. The amplitude is the linear distance from the unstretched equilibrium position.
3. The restoring force for a horizontal mass-spring system is the elastic force from the spring(s);
4. The restoring force for a vertical mass – spring system is the elastic force and gravity.
5. T = 2π√
𝑚
𝑘
where m is in [kg], k is the spring constant [N/m] and T is in [s].
This is neat: Mass swinging and bouncing on a spring
C. Torsional Pendulum
1. The rotating rod on the right is a good example of a torsional pendulum.
2. The amplitude is the angular distance from the untwisted equilibrium position.
3. The restoring force is the torsion from the twisted string, spring, rope, etc.
4. We will not calculate periods for these.
Damping
1. a damping force causes an oscillator to lose Amplitude
2.
the system will still undergo SHM (if the damping force is not too large), but with a slightly
lower frequency
3.
usually caused by friction (internal or external)
Underdamped
a. causes a loss in A, but the T and f remain constant until the oscillator stops
b.
all real world oscillators experience friction, so they're all underdamped
c.
get a swing swinging, and stop driving the system with your legs or a push – the swing
will continue swinging with smaller and smaller A, but still with well defined frequency.
d. This graph shows the amplitude (height of the wave) diminishing over time, but the
period remains constant
Critically Damped
a. enough force to stop the oscillator from going back through its equilibrium position
half an oscillation
i.e.
b. Shock absorbers are a good example. After hitting a bump, the car goes up and back to
normal without continually oscillating – which is a good thing for drivers, passengers, etc.
c.
The graph shows the oscillator returning to equilibrium and not passing back through
Overdamped
a.
after displacement, the system slowly goes back to equilibrium
b. a good example is a door with a pneumatic stop (those air chamber thingies that slowly
let the door back)
c.
another good example is oleo struts on airplanes (shocks). The weight of the plane really
depresses the strut on landing, then it slowly pushes the plane back to normal.
d. these systems do not have well defined frequencies or periods
Something to mess around with
IMPORTANT DEFINITION: Natural Frequency (Resonant Frequency): all systems
have a certain frequency at which they oscillate, or vibrate. That frequency is its natural or
resonant frequency.
1. For SHM, the natural frequency is easy to calculate.
2. For more complicated systems, the natural frequency is apparent when you hear or
see it vibrate
3. Most systems have several natural frequencies
IMPORTANT DEFINITION: Resonance: an oscillating system forces another system to
oscillate (with the same frequency)
1. the frequency of the oscillators must be integer multiples of each other’s resonant
frequencies
2. f(1) = nf(2) where n = 1, 2, 3, ...
3. The demo in class is a good example.
4. Lots more on this in wave mechanics and sound.
Forced Oscillations: A driving force applied with a consistent frequency causes a system
to vibrate with the frequency of the force
1. If the driving force matches the resonant frequency of the system, the A will
increase dramatically with less force, less energy expended.
2. Watch MythBusters. They did two different episodes with oscillators that tried to
shake bridges. One was a suspended bridge with marching feet; the other was a
large metal framed bridge like the one in East Troy. This is their test of the actuator
on a steel beam.
3. A person periodically pushed a person on the swing. If the force is timed to match
the resonant frequency, the swing will go higher and higher. If they don’t time the
force right, then they are just silly.
4. If a hold a slinky, and force it up and down, it will oscillate at the frequency that my
hand delivers the force .
5. A motor drives the torsional oscillator, which forces your clothes down into the water.
.