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Plan for Quiz 5 to be Tue.Oct.22 thR.Oct.24
... plan for Exam 3 to be Tue.Nov.12

Topic 5 (Oscillations & Vibrations)

where things are disturbed from a stable equilibrium

objects usually arrange themselves to be in an equilibrium configuration , where Σ F = 0 ... so they can stay there

If pushed away from the equilibrium point (uphill), the hillside applies a Force toward the stable equilibrium place (hence "stable")
. . . the ball is accelerated toward the equilibrium (bottom), but is going fast there, so it overshoots and goes up the other (−x) hillside
      slowing as it goes uphill until it runs out of KE ... at the turning point ... one Amplitude distance from equilibrium
. . . gradient of the PE will push the ball toward equilibrium again (if it is a stable one ! )
      of course, then it will again overshoot the bottom , slowing as it rises up the (+) sidehill, to stop at its maximum distance from eq.
=> the distance from the equilibrium location to the turning point is called the Amplitude of its oscillations ... abbreviated A , measured in meters.
=> the Time Period   T   of this oscillation is from (+)x_max thru 0 to (−)x_min , thru 0 to (+)x_max again ... measured in seconds/cycle.
=> the oscillation frequency is the rate of oscillating ... f = 1/T , in cycles/ second   (= "Hertz" , after Heinrich, who demonstrated radio oscillators in the 1880's)

the Force is stronger (steeper hillside) where farther from the bottom , so the ball accelerates quicker there
      with a uniform curvature bowl (parabola shape), the faster average speed exactly compensates for the extra travel distance.
      the time for one oscillation is the same , no matter how far from equilibrium its x_max gets.
      . . . typical springs are like this ... they follow Hooke's Law (named for Robert, who taught Newton about springs).
. . . rapid oscillation frequency occurs for low-mass objects moving in tight curvature PE (small bowl)
=> 2 π f = √ k/m
      so it takes a long Time for a massive ("sluggish") object to oscillate in a nearly flat PE dish (limp spring) .
      tighter PE curvature   large k   means the mass is attached to a stiff spring ... either thick material or short .

The oscillator has only PE at its extreme positions (where it has zero KE) , but has that same amount of KE as it goes thru equilibrium (where PE=0).
      . . . part-way up the hillside , the oscillator will have half PE and half KE ... the total mechanical E is constant!
      a Hooke's Law (quadratic PE) oscillator travels in a cosine curve as a function of time, with velocity that is a −sine curve;
      there is only one frequency, so that kind of motion is called a "Simple Harmonic" Oscillation.
=> averaged over 1 cycle time, the oscillator has ½ its Energy as KE , and ½ its Energy as PE .

Pendulum : mass on a string in gravity . . . the restoring Force depends on the mass, so the ratio Force/inertia does NOT .
      PE curvature depends on the string length : shorter string curves tighter
=> repeat period = 2 π √ L/g   . . . mass on a 1 meter-long string has a 2 second period
. . . but mass needs a 4 meter string to have a 4 second period.
real pendulums take a few % longer to do "large angle" oscillations (±60° or ±80°) than small-angle oscillations (±20°)

Floating objects bob up and down much like Simple Harmonic Oscillators
. . . when too high in the water, their weight is stronger than the bouyant Force, so net (total) Force is downward
. . . when too low in the water, the bouyant Force is stronger than the gravity Force, so the net Force is up
. . . but motion thru the water "loses" Energy as water's viscous Force resists motion, doing negative Work to it
=> the oscillations are damped (die out gradually ... less Work done as Amplitude decreases)

Moon is sometimes closer to Earth than average, sometimes farther away.
. . . it appears to be oscillating around its average distance from us
      6% closer than average, then 6% farther than average ...
. . . 405 504 km at maximum, 363 396 km at minimum ... the amplitude of its oscillations
      is 6 054 km (half of the apogee − perigee difference)
=> that same phenomena is explained by saying that its "orbit" around Earth is elliptical

Resonance (Frequency and Timing) ; Damping

Suppose a swing has a 3-second Time Period : starting from rest,
. . . if you push every 3 seconds, during the downward part of the swing, each push adds Energy to the swing
      the swing height (above the bottom) would increase maybe 0.1 m each push
. . . if you push for the entire 1½ second of its forward motion (imagine push near the pivot)
      you could double each gain ... to add 0.2 m height each oscillation

Gentlest Force does the most Work, by pushing (forward) during forward motion,
. . . but pulling (backward) during backward motion ... match the entire cycle.
      that could add 0.4 m height to each swing ... 3.2 m high after 8 swings?
      except that friction does more (neg) Work if it travels farther.
. . . eventually each push just replaces the Energy "lost" to friction
      the Amplitude is not infinite even if your Force timing resonates with the swing.
=> Energy losses from an oscillator are called "damping": friction, air viscosity & drag, sound!

Suppose the swing (above) is frictionless, but you push every 3½ seconds instead of every 3 seconds.
. . . your first push (starting at time 0) gives it 0.2 m height : that swing ends at 3 s with A1 = 0.20 m height.
. . . your next push starts ½ s late; ok, the first 1 s of push is during the swing's + motion
      but the last ½ second of the push "undoes" the ½ s of Work just before it
      ... because the swing's motion is negative then - so only adds .067 m [A2 = 0.267 m]
. . . your push starting at 7 seconds has only the first ½ s doing positive Work, and 1 s doing negative
      so it subtracts .067 m of height by the end of that swing [A3 = 0.20 m]
. . . a push starting at 10½ seconds does all negative Work, removing all the swing's Energy [A4 = 0 m]
=> timing is crucial for sustaining resonance !
      frequency off by 16% here (½ s / 3 s) ... maximum amplitude was only 1½ × the first one.
. . . OTOH, narrowness of resonances means that they can be avoided by design
      (rattles and vibrations in a car silenced by a "small" speed change)
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