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How was Midterm 2 for you? A) Too hard and/or long -­‐ no fair! B) Hard, but fair C) Long, but fair D) Seemed reasonable. E) VERY reasonable, thanks!! 1
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Something about Gauss’ law
Or Density = rho * V!
2
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Consider a super ball which bounces up and down on
super concrete. After the ball is dropped from an initial
height h, it bounces with no dissipation and executes
an infinite number of bounces back to height h.
Is the motion of the ball in z simple harmonic motion?
A)  yes
B)  no
C)  ???
5 Quick Review aNer the break If a spring is pulled away its equilibrium point to x>0 and released from rest, the plots shown below correspond to I A)  I: velocity, II:posiVon, III:acceleraVon II B) I: posiVon, II:velocity, III:acceleraVon C)  I: acceleraVon, II:velocity, III:posiVon D) I: velocity, II:acceleraVon, III:posiVon III E)  I do not have enough informaVon 3
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Important concepts Angular frequency ω =
Frequency: f =
1
T
2π
k
=
T
m
PosiVon: x(t ) = A cos(ωt + δ )
Velocity: v(t ) = − Aω sin(ωt + δ )
2
AcceleraVon: a(t ) = −ω x(t )
Energy: E =
2
mv 1 2 1 2
+ kx = kA
2
2
2
No fric0on implies conserva0on of mechanical energy Damped OscillaVons In most physical situaVons, there are nonconservaVve forces of some sort, which will tend to decrease the amplitude of the oscillaVon. In many cases (e.g. viscous flow) the damping force is proporVonal to the speed: Fdrag = −bv 2β = b / m
F = mx = −kx − bx
x + 2βx + ω0 2 x = 0
4
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Underdamped OscillaVons An underdamped oscilla/on with b<w0 : Note that damping reduces the oscillaVon frequency. Overdamped OscillaVons An overdamped oscilla/on with b>w0 : 5
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CriVcally damped OscillaVons A criVcally damped oscilla/on with b=w0 : Damped oscillaVons 6
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This class Driven OscillaVons & Resonance 1. ParVcular and homogeneous soluVon 2. Response to periodic driving forces: amplitude and phase. 3. Tutorial 4.  Resonance: concept, Q factor 14 Driven oscillators and Resonance: phenomena occur widely in
natural and in technological applications:
Emission & absorption of light
Lasers
Tuning of radio and television sets
Mobile phones
Microwave communications
Machine, building and bridge design
Musical instruments
Medicine
– nuclear magnetic resonance
magnetic resonance imaging
– x-rays
– hearing
Nuclear magne0c resonance scan CP 465 7
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What is the parVcular soluVon to the equaVon y’’+4y’-­‐12y=5 ? A) 
B) 
C) 
D) 
y=e5t y=-­‐5/12 y=-­‐a 5/12, a determined by iniVal condiVons y= a e2t +b e-­‐6t +5 Consider the following equaVon x’’+16x=9 sin(5 t), x(0)=0, x’(0)=0 The soluVon is given by A. x(t)=c1 cos[4 t]+ c2 sin[4t]-­‐ sin(5 t) B. x(t)=5/4 sin[4t]-­‐ sin(5 t) C. x(t)= cos[4t]-­‐ cos(5 t) D. I do not know 8
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A person is pushing and pulling on a pendulum with a long rod. Rightward is the posiVve direcVon for both the force on the pendulum and the posiVon x of the pendulum mass To maximize the amplitude of the pendulum, how should the phase of the driving force be related to the phase of the posi0on of the pendulum? 17 What is the most general form of the solution of
the ODE u’’+4u=et ?
A) u=C1e2t + C2e-2t + C3et
B)  u=Acos(2t-δ) + C3et
C) u=C1e2t + C2e-2t + (1/5)et
D) u=Acos(2t-δ) + (1/5)et
E)  Something else!???
18
9
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The phase angle d in this triangle is B A A. ⎛B⎞
⎝ A⎠
δ = arctan ⎜ ⎟
B. δ = arc cot⎛⎜ A ⎞⎟
⎝B⎠
⎛ A⎞
C. δ = arctan⎜ B ⎟ D. More than one is correct ⎝
⎠
D. None of these are correct Consider the general soluVon for an underdamped, driven oscillator: Which term dominates for large t? x(t ) = C1e−β t e
+ β 2 −ω02 t
term A + C2e−β t e
− β 2 −ω02 t
term B + A cos(ωt − δ )
term C D) Depends on the parVcular values of the constants Challenge quesVons: Which term(s) malers most at small t? Which term “goes away” first? 20 10
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Consider the amplitude 2
f0
A = 2
2 2
(ω0 − ω ) + 4β 2ω 2
2
In the limit as ω goes to infinity, A A. Goes to zero B. Approaches a constant C. Goes to infinity D. I don’t know! Consider the amplitude 2
f0
A = 2
2 2
(ω0 − ω ) + 4β 2ω 2
2
In the limit of no damping, A A. Goes to zero B. Approaches a constant C. Goes to infinity D. I don’t know! 11
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What is the shape of A=
1
(ω
2
0
− ω 2 ) + 4β 2ω 2
2
d 2Q
dQ Q
L 2 +R
+ =0
dt
dt C
Given the differenVal equaVon for an RLC circuit, which quanVty is analagous to the damping term in a mechanical oscillator? A)  R, resistance B)  L, inductance C)  C, capacitance Challenge quesVon: Which quanVty is analogous to the mass 24 12
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d 2Q
dQ Q
L 2 +R
+ =0
dt
dt C
What is ω0? A) 
B) 
C) 
D) 
E) 
C 1/C 1/Sqrt[C] 1/LC 1/Sqrt[LC] 25 Driven OscillaVons & Resonance If the driving frequency is close to the natural frequency, the amplitude can become quite large, especially if the damping is small. This is called resonance. Small damping: sharp peak Large damping: broad peak March 11, 2011 Physics 114A -­‐ Lecture 32 26/32 13
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Driven Oscilla0ons & Resonance ω0 =
Δω
ω0
=
1
Q
k
m
The Q-­‐factor characterizes the sharpness of the peak Can you break a wine glass with a human voice if the person sings at precisely the resonance frequency of the glass? A) Sure. I could do it B)  A highly trained opera singer might be able to do it. C)  No. Humans can’t possibly sing loudly enough or precisely enough at the right frequency. This is just an urban legend. 14