421 Oscillations 1 Are oscillations ubiquitous or are

421: Oscillations 1

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

PH 421: Homework 30%; Laboratory reports 40%; Final 30%. All lab reports will be submitted in class. Mon 10 Energy Diagrams 17 Free damped motion Pendulum lab due! 24 Fourier Series 1 Fourier coefficients & transform ~ November 2014 ~ Wed 11 12 Lab & Discussion: the -HW 1(1, 2) due Pendulum -Bring lab graphs to Upload Data class -Lab analysis 18 19 Lab: LCR harmonics -HW 2(1, 2) due Forced motion & - Upload data resonances - LCR lab analysis Tue 25 Research in the Physics; intro to senior thesis 2 The Fourier transform HW 3(1, 2) due 26 -LCR lab due Thu 13 Simple Harmonic Motion Fri 14 -HW 1 due Free oscillatory motion 20 Forced motion & resonances - LCR circuit 27 3 4 The Fourier transform Loose ends and review -demo lab HW 3 due 21 -HW 2 due Multiple Driving Frequencies 28 5 Optional review 3

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

PH 421: Oscillations – lecture 1 Reading: Taylor 4. 6 (Thornton and Marion 2. 6) (Knight 10. 7) 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? But what if v is not constant? Separation of x and t variables! 8

The case of a conservative force Suppose total energy is CONSTANT (we have to know it, or be able to find out what it is) 9

Example: U(x) = ½ kx 2 , the harmonic oscillator B x. L x 0 x. R Classical turning points 10

Symmetry - time to go there is the same as time to go back (no damping) x. L x 0 x. R SHO - symmetry about x 0 x. L -> x 0 same time as for x 0 -> x. R 11

SHO - do we get what we expect? x. L=-A 0 x 0 x=0 x. R=A Another way to specify E is via the amplitude A Independent of A! 12

x. L=-A x 0 You have seen this before in intro PH, but you didn't derive it this way. x. R=A 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.

B x. L x 0 “Everything is a SHO!” x. R 16

Equivalent angular version? 17

Integrate both sides E and U(q) 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 18 web page)

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, q Your lab example is a plane pendulum. You will have to generalize: 19 what length does L represent in this case?

q L mg 20

Plot these to compare to SHO to pendulum…… 21

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