1 421: Oscillations

2 Are oscillations ubiquitous or are they merely a paradigm? Superposition of brain neuron activity

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 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.

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

Intro to Research in Physics Wk 3: devoted largely to introducing the senior thesis research/writing requirement If you’re 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

Reading:Taylor 4.6 (Thornton and Marion 2.6) (Knight 10.7) PH421: Oscillations – lecture 1 6

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

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

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

Classical turning points xLxL xRxR B x0x0 Example: U(x) = ½ kx 2, the harmonic oscillator 10

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

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 Yhu/Animations/sho.html 12

x L =-Ax R =Ax0x0 You have seen this before in intro PH, but you didn't derive it this way. 13

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

Period increases because v(x) is smaller at every x (why?) in the trajectory. Effect is magnified for larger amplitudes.

xLxL xRxR B x0x0 “Everything is a SHO!” 16

Equivalent angular version? 17

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

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

20  mg L

Plot these to compare to SHO to pendulum…… 21

Nyquist theorem; sampling rate is critical if the sampling rate < 1/(2T), results cannot be interpreted Time (s) Displacement (degrees)