Physics 1 Topic 7: Oscillations Thru Equilibrium
Systems are often found "near" their equilibrium conditions
... Forces nearly cancel , so accelerations are small
If the thing's PE(x) graph is a local maximum (hilltop) , the object will move slowly at that unstable equilibrium
. . . as the object gets further from equilibrium , the KE increases, since the Force (−δPE(x) / δx) points away from equilibrium
the object is not likely to return to that hilltop ... no repeating.
If the object is at a PE(x) local minimum ("valley","well",bowl) , the object will be restored to equilibrium
. . . (pushed back toward it) . . . whenever displaced from equilibrium
by the "inward" gradiant (location-slope) of the PE graph ... [J/m] = [N] ... at its turn-around point.
⇒ this gradient is called a restoring Force .
The object gains KE and momentum as it re-approaches the valley bottom
. . . this momentum will carry it past equilibrium , over-shooting all the way to a turn-around point on the other side.
. . . after being restored to equilibrium from that side, it overshoots again and returns to its starting condition.
⇒ Time taken to repeat a "motion cycle" is called the Repeat Time T or the Time period T .
The amount of time it takes to complete one cycle is called the Repeat Time or the oscillation Time Period [second/cycle] , symbol T is not Tension !
. . . alternatively , the number of cycles that are completed in a time interval is called the oscillation frequency (cycle frequency, counting frequency) f [cycle/sec] .
⇒ T [s/cycle] = 1/f . . . f [cycle/sec] = 1/T . . . the [cycle/s] unit is also called a Hertz [Hz] .
Distances are measured from the equilibrium location ... the maximum distance is called the Amplitude (abbreviated A).
. . . we can set up a PE well by putting a ramp on each side of a cart ;
if the ramps have different slope, the Amplitude on the steeper ramp will be shorter (distance) ... same height.
the cart will have 2 different constant accelerations (or 3, if there's flat space with a=0 between the ramps)
The repeat time will depend on how far the cart travels along each ramp: A = ½ a T/4 ²
⇒ T = 2 √(2A/a) right + 2 √(2A/a) left .
... it's not very pretty having large amplitude motion take longer than small-amplitude motion.
In order for the oscillation Time Period to be the same for large-Amplitude motion as for small-Amplitude motion ,
the Restoring Force must become much stronger at large displacements from equilibrium .
. . . this means that the slope of the PE well . . . the gradient of PE(x) ... (ΔPE / Δx ) ... must be steeper at large distances.
typical Hookes-Law springs are just what we need so that all amplitudes have the same repeat-time.
⇒ PE = ½ k x² has the appropriate steepness function ... F = − k x ... k is the spring stiffness.
Some repeating phenomena have particularly smooth motion ... "sinusoidal" or "harmonic" oscillations
. . . if the motion follows a single sine curve time-wise, it is called "simple" harmonic motion.
ωosc = 2 π [radians/cycle] · f [cycles/s] => ω [rad/s] ... along the oscillation time-cycle in the trig function argument
. . . the smoothest motion occurs when the PE well has uniform kurvature ks = − ΔF / Δs ... with a parabolic PE .
We measure the PE relative to the equilibrium location's PE ... so PE(0) = 0 , as seen above.
. . . If Energy is conserved (ignoring friction) , the KE maximum (at the PE bottom)
will equal the PE at the turning point (at the extreme A) :
½ k A ² = ½ m v ² = ½ m (A ω)² .
⇒ ωosc = √(k/m) ...
We like ω because Nature does ... descriptions of simple harmonic motion look clean and simple using ω !
x(t) = A cos (ω t + φ) . . . <=> . . . x(t) = A cos (2 π t/T + φ) ;
v(t) = − A ω sin (ω t + φ) . . . <=> . . . v(t) = − A (2 π/T) sin (2 π t/T + φ) ;
a(t) = − A ω² cos (ω t + φ) . . . <=> . . . a(t) = − A(2 π/T)² cos ( 2 π t/T + φ) .
"Smooth" oscillations like those above have PE(x) graphs that are parabolas ... like a typical spring : PE = ½ks²
. . . the spring stiffness is the "kurvature" of the PE(x) graph ... the slope of the slope (second space derivative , as d²/dx² PE) ... is konstant .
(this is location-slope = "gradient" ... a time-slope = "rate" analogy is acceleration : parabolic x(t) => steadily increasing velocity => constant a = the parabola's curvature )
All PE(x) graphs near stable equilibrium locations are approximately parabolic (curve upward)
. . . so the most general model of things that oscillate is , at its core , about inertia in a parabolic PE well
... usually called a mass-on-spring oscillator ... with ω = √k/m .
notice that there are TWO KE maximums during each cycle (since −v at equilibrium also has positive KE),
. . . and TWO PE maximums (since the −A extreme also has positive PE).
A simple pendulum is a mass on a string that swings through the "straight-down" orientation .
The mass traces out an arc - part of a circle - but if the maximum angle isn't very far from vertical ,
. . . that arc (and the PE(x) graph for the mass ... from mgh) nearly matches a parabola
with ; mg sinθ is the part of the Force parallel to v,
. . . and the distance is has moved from equilibrium is Lθ , so => k = −F/s = mg sinθ/Lθ ≈ mg/L .
. . . plugging this small-angle approximation for k , in the "general model" , gives ω = √g/L .
A physical pendulum is an extended mass that can swing around a pivot, left and right across the "straight-down" orientation .
If the center-of-mass is not exactly under the pivot, but is some angle from that, gravity Force pulling on the c.o.m. causes a torque
which angularly accelerates the object's rotational Inertia ... downward", toward equilibrium.
. . . the kurvature of the angular PE as a function of angle [in radians] ... is the slope of the Torque function .
. . . the object's rotational Inertia makes its rotational angular acceleration "sluggish".
the small-angle approximation for the oscillation angular velocity is (by rotational analogy to the linear formula)
=> ω ≈ √k/m = √( (dτ/dθ) / I ).
we should check that this formula gives the same frequency for a simple pendulum as the other formula.
. . . I = m L² . . . τ = m g L sin θ ≈ m g L θ => dτ/dθ = m g L .
To put Energy into an oscillating system, you need to push and/or pull with appropriate timing.
your Force should do positive Work to the mass, as it moves.
. . . The most effective way to do this is to push forward while it is already moving forward,
and pull backward while it is moving backward
⇒ same frequency as its natural motion ... then your Force "resonates" with the oscillator.
Most real oscillating systems have some friction which decreases its Energy.
. . . sliding friction removes Energy based on its travel distance (μN)(4A),
with a small friction coefficient, the Amplitude will decrease by a small amount each cycle:
⇒ E − ΔE = ½ k A² − 4μN A . . . ΔA is complicated.
. . . viscous (fluid) resistance depends on the velocity, so does work proportional to (b ωA)(4A)
with small damping b, the Amplitude decreases by the same small percentage each cycle
⇒ ΔA = −f A . . . A decays exponentially to zero ... takes infinite time to die completely.
