2 Are oscillations ubiquitous or are they merely a paradigm?
Superposition of brain neuron activity
Slide 4
3 ~ November 2014 ~ MonTueWedThuFri 10 Energy Diagrams 11 Lab
& Discussion: the Pendulum Upload Data 12 -HW1(1,2) due -Bring
lab graphs to class -Lab analysis 13 Simple Harmonic Motion 14 -HW1
due Free oscillatory motion 17 Free damped motion Pendulum lab due!
18 Lab: LCR harmonics Forced motion & resonances 19 -HW2(1,2)
due - Upload data - LCR lab analysis 20 Forced motion &
resonances - LCR circuit 21 -HW2 due Multiple Driving Frequencies
24 Fourier Series 25 Research in the Physics; intro to senior
thesis 26 -LCR lab due 2728 1 Fourier coefficients & transform
2 The Fourier transform HW3(1,2) due 3 The Fourier transform -demo
lab 4 Loose ends and review HW3 due 5 Optional review PH421:
Homework 30%; Laboratory reports 40%; Final 30%. All lab reports
will be submitted in class.
Slide 5
Intro to Formal Technical Writing Two formal lab reports (40%)
are required. Good technical writing is very similar to writing an
essay with sub-heading. We want to hear a convincing story, not a
shopping list of everything you did. Check out course web-site
4
Slide 6
Intro to Research in Physics Wk 3: devoted largely to
introducing the senior thesis research/writing requirement If youre
thinking about grad school, med school, etc. and have not
started/planned out research opportunities you are already behind
the competition. Start now! -due. Nov. 17: URSA-ENGAGE Research
opportunity for sophomores/transfers -due Feb. Department SURE
Science scholarship -due (very soon) external competitions, REUs,
etc. 5
Slide 7
Reading:Taylor 4.6 (Thornton and Marion 2.6) (Knight 10.7)
PH421: Oscillations lecture 1 6
Slide 8
Goals for the pendulum module: (1) CALCULATE the period of
oscillation if we know the potential energy; specific example is
the pendulum (2) MEASURE the period of oscillation as a function of
oscillation amplitude (3) COMPARE the measured period to models
that make different assumptions about the potential (4) PRESENT the
data and a discussion of the models in a coherent form consistent
with the norms in physics writing (5) CALCULATE the (approximate)
motion of a pendulum by solving Newton's F=ma equation 7
Slide 9
How do you calculate how long it takes to get from one point to
another? Separation of x and t variables! But what if v is not
constant? 8
Slide 10
Suppose total energy is CONSTANT (we have to know it, or be
able to find out what it is) The case of a conservative force
9
Slide 11
Classical turning points xLxL xRxR B x0x0 Example: U(x) = kx 2,
the harmonic oscillator 10
Slide 12
xLxL xRxR x0x0 Symmetry - time to go there is the same as time
to go back (no damping) SHO - symmetry about x 0 x L -> x 0 same
time as for x 0 -> x R 11
Slide 13
x L =-Ax R =A x0x0 SHO - do we get what we expect? Another way
to specify E is via the amplitude A Independent of A! x0=0x0=0
http://www.wellesley.edu/Physics/ Yhu/Animations/sho.html 12
Slide 14
x L =-Ax R =Ax0x0 You have seen this before in intro PH, but
you didn't derive it this way. 13
Slide 15
Period of SHO is INDEPENDENT OF AMPLITUDE Why is this
surprising or interesting? As A increases, the distance and
velocity change. How does this affect the period for ANY potential?
A increases -> further to travel -> distance increases ->
period increases A increases ->more energy -> velocity
increases -> period decreases Which one wins, or is it a tie?
14
Slide 16
Period increases because v(x) is smaller at every x (why?) in
the trajectory. Effect is magnified for larger amplitudes.
Slide 17
xLxL xRxR B x0x0 Everything is a SHO! 16
Slide 18
Equivalent angular version? 17
Slide 19
Integrate both sides E and U( ) are known - put them in
Resulting integral do approximately by hand using series expansion
(pendulum period worksheet on web page) OR do numerically with
Mathematica (notebook on web page) 18
Slide 20
Look at example of simple pendulum (point mass on massless
string). This is still a 1-dimensional problem in the sense that
the motion is specified by one variable, Your lab example is a
plane pendulum. You will have to generalize: what length does L
represent in this case? 19
Slide 21
20 mg L
Slide 22
Plot these to compare to SHO to pendulum 21
Slide 23
Nyquist theorem; sampling rate is critical if the sampling rate
< 1/(2T), results cannot be interpreted Time (s) Displacement
(degrees)
