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Intro Physics I (Phy.211) 2010 Fall
Unit 3 ; Topic Eight - Oscillations
Plan for Unit 3 :
-Topic 7:
Torque by Force & Angular Kinematics : Ch.12 ; Ch.14 § 1,2
. . . (HW due Tue.Oct.26)
Torque & Angular Momentum change : Ch.13
. . . (HW due Thr.Oct.28)
Quiz Mon.Nov.01 Wed.Nov.03
-Topic 8:
Oscillations : Ch.15 § 1,2,3
. . . (HW due Tue.Nov.02)
Pendulums & coupled Osc's : Ch.15 § 5 , 4
. . . (HW due Fri.Nov.05)
Quiz Mon.Nov.08
-Topic 9:
Wave Propagation : Ch.16 § 1,2 ; Ch.17 § 1,2,3
. . . (HW due Tue.Nov.09)
Wave Interference : Ch.16 § 3,4 ; Ch.17 § 4,5
. . . (HW due Thr.Nov.11)
- plan for Exam 3 Wed.Nov.17
Home-Work PRACTICE QUESTIONS for Topic 8 ...( NOT for grading ...):
(numbers refer to Ohanian Physics for Engineers,3rd ed.)
Kinematics : ch.15 prob. 1,2,5,7,14
Simple Harmonic F : ch.15 prob. 17,19,20,24,25,26,29
Energies : ch.15 prob. 33,34,35,36,37,41
gravity Pendulums : ch.15 prob. 43,47,48
Physical Pendulums : ch.15 prob. 53,55,57,63,65,67,69 ; 70,72,73,74
resonance & Δ E: ch.15 prob. 77,79,80,83,83
Some Exam 3 Practice questions available at mu-online (WebCT)
Homework 13 (from Fri.Oct.29 - Wed.Nov.03 ... classes 38 - 41) - due Fri.Nov.05
- A ¼ [kg] mass on a spring is pulled −0.4[m] from equilibrium, released at t=0, has v=2[m/s] (z) as it crosses equilibrium.
a) what Work was done on the system by the hand-pull from equilibrium? (hint: KE)
b) at what time did the mass first cross equilibrium?
c) what is its acceleration (vector!) at 2× tequilib. ? { where is it then? }
d) graph the system's KE from 0[s] for the first 1½ oscillations; then graph PE (label these clearly!)
- A crystal oscillator (for timing) expands and contracts 477 million times each second, if the moving mass is .001[kg] .
(a) compute the angular frequency for this oscillation.
(b) calculate the effective "spring" stiffness for a crystal with this Area
(c) Suppose the oscillator contains 9[J] mechanical Energy ... what is the amplitude of motion?
(d) if the mass increases by 0.0001 [kg], does the frequency increase or decrease? ... by what %?
- We take a 0.4[kg] heavy meterstick and suspend it from a small hole at 0.333[m]-mark, on a (frictionless) pivot.
a) what's the frequency (f or ω) for a simple pendulum with length 0.167[m]?
. . . explain how the simple question pendulum question is relevant to the physical pendulum situation.
b) What angular frequency would the small-angle formula for a physical pendulum predict for this set-up?
c) If the c.o.m. amplitude is 0.2618[m], with oscillation ωosc from (b) , what maximum velocity would the stick c.o.m. have?
d) Use c.o.m. speed (from c) to get rotational speed ωrot , then obtain the maximum KE.
. . . how does that compare with the initial PE that the stick had (at 90°)?
- The balance wheel in an antique pocketwatch oscillates torsionally ; from equilibrium, it spins clockwise 3 turns in 1/8 [sec], then back to equililibrium in the next 1/8 [s].
The 10[mm] radius (hoop-shaped) wheel has a mass of about 0.3[gram] (NOT kg].
(a) how far does the outer surface travel from the equilibrium orientation? ... find the maximum Δθrotation [rad]
(b) what's the average speed of this balance-wheel outer surface, spinning from −3 rev to +3 rev ... what's the average ωrotation
(c) what is the oscillation angular frequency ;ωoscillat? ... use with amplitude to get flywheel maximum speed
(d) what's the maximum rotational acceleration for the wheel? what is the maximum rotational torque applied to it? where is it then?
Topic 1 Reminders :
(to topic 1 summary)
USE UNITS all thru your answer ... don't just append them onto the final number !
Put symbols on a sketch, as you read . . . use meaningful phrases : "v_match" , "catch" , "ground" "turn-around"
. . . show process spans as brackets or vector arrows , with words like "average" , "losing" , "gaining" , "t=0→2" ...
Write statements as symbols first ... each statement has a SUBJECT ... manipulate symbols before plugging numbers.
. . . keep track of adjectives such as "initial" , "average" , "final" , "stopped" . . . "change"
Topic 2 Reminders :
(to topic 2 summary)
subscripts for "total" or "system" , "part A" ...
keep track of the SOURCE for the external Forces applied to the (passive) object
Recognize whether you are predicting based on theory . . . reasoning from causes to their effects ... (what should it do?)
or whether you are deducing from observation . . . (what does this imply about it?)
Get un-stuck (often) by wondering "why isn't it the same as it used to be?"
Topic 3 Reminders :
it's the Sum of external Forces that cause an object's mass to accelerate subscripts for "total" or "system" , "part A" ...
once you separate vectors into components, keep each component separate from the others!
ΣF = ma , for each component separately.
Topic 4 Reminders :
Name each Force by the source of that Force ... its cause ... a function of the environment.
spring Force : Fspring = − k s ... opposite the stretch vector .
Pressure Force : Fspring = P A generally ... in a fluid , δP = ρ g δh .
Gravity : every mass in the universe contributes to the gravity field g at any place of interest.
... add "nearby" contributions as vectors ... gby M = G M / r²M-to-point (toward M) . . . then use Fgravity, on m = mg ; NOTICE : m ≠ M !
Topic 5 Reminders :
Energy is conserved quantity : Σ Ebefore + Σ Wby F's not in PE list = Σ Eafter .
KE = ½ m v² = ½ p·v = ½ p²/m , in any of these equivalent forms .
Each Force either does Work during motion , or has an associated PE function ... not both .
=> Wby F = F·Δs .
=> − dU / dr = Fr , the component along that direction (r)
PEgravity,local = m g h , with h measured upward (z-direction) from a "zero-height reference" point
PEelastic = ½ k s² . . . stretch s must be measured from its relaxed length
PEPressure = P·V ; it is okay to measure from a non-zero reference pressure , but be consistent !
PEGravitation = − m GM / r .
There IS NO PE for Friction Force ... must be explicitly included as a "Work by Forces not in the PE list"
Topic 6 Reminders :
Some Energy forms are somewhat complicated to calculate, at a fundamental level, such as Chemical PE or Nuclear PE.
. . . they can be treated as experimentally-measured quantities which are "released" (transformed) during some conversion process (i.e, burning).
more material holds more Energy ; it is the PE density (PE/Volume) or the specific PE (PE/mass) that is measured.
efficiency is the fraction (or %) of the input Energy that is transformed into the "desired form".
. . . because total Energy is consserved, the sum of efficiencies in all output forms must be 100%.
Power is the rate that Energy is transformed (or transfered) into some other type (or to some other object)
=> P = ΔE / Δt , reported in [Watt] = [J/s] .
Because KE goes as v² = vx² + vy² + vz² , motion along each component adds in an independent manner .
. . . in particular, radial and angular motion components can be treated separately for objects in orbit
=> L = r × p is conserved then , so that KEangular = L²/2mr² is a useful formula .
Topic 7 Reminders :
just as in straight-line motion, a thing's angular momentum tends to stay the same.
Most angular motion formulas and equations are obtained by directly replacing each linear quantity by its angular sibling
. . . angular location, or angular position, angular orientation φ [radians] = (arc length) s/r <=> x, y, or z .
. . . angular velocity ω [radian/sec] = dφ/dt = v/r is analogous to linear velocity
. . . angular acceleration α [radian/s²] = dω/dt = a/r is analogous to linear acceleration .
. . . angular inertia I [kg m2] = Σ m r² is analogous to mass
=> Angular momentum L = r × p ... rite rist along r, pingers point to p , thumb shows L can write as L = I ω
. . . r × F = Torque τ . . . right-hand rist along r , fingers flip thru angle φ till point along F ; Thumb points ( | ) along Torque
. . . Torque is the angular analogy to (linear) Force => Σ τ = dL/dt . . . sometimes useful to write as Σ τ = I α
Work (a scalar) is its own analogy ... Wrot = τ ·Δθ . . . Power P = τ · ω
Kinetic Energy is its own analogy ... KErot = ½ I ω² = ½ L ω = L²/2I
Topic 8 Summary :
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 it gets further from equilibrium , the PE(x) graph gets steeper , so the Force away from equilibrium becomes stronger
. . . (hence accelerates quickly away) .
. . . if the thing is at a PE local minimum (valley,bowl,well) , the object will be restored to equilibrium
. . . (pushed back toward it) . . . whenever displaced from equilibrium
by the "inward" slope of the PE graph ... [J/m] = [N] , so this is called a restoring Force .
the object's inertia will carry it past equilibrium , to over-shoot onto the other side
. . . after being restored to equilibrium from that side, it overshoots again and returns to its starting condition.
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 . . . [cycle/s] is also called Hertz [Hz] .
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 (d/dx) of PE(x) . . . must become steeper at larger distances.
=> PE = ½ k x² has the appropriate steepness function so that all amplitudes take the same time to repeat .
Some repeating phenomena have particularly smooth motion ... "sinusoidal" or "harmonic" oscillations
. . . if one frequency describes the motion adequately , it is "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 = − dF / ds ... parabolic PE .
=> 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 .
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 topic 9 summary)
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