1 421: Oscillations 2 Are oscillations ubiquitous or are they merely a paradigm? Superposition of...
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Transcript of 1 421: Oscillations 2 Are oscillations ubiquitous or are they merely a paradigm? Superposition of...
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- 1 421: Oscillations
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- 2 Are oscillations ubiquitous or are they merely a paradigm? Superposition of brain neuron activity
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- 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.
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- 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
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- 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
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- Reading:Taylor 4.6 (Thornton and Marion 2.6) (Knight 10.7) PH421: Oscillations lecture 1 6
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- 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
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- 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
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- 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
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- Classical turning points xLxL xRxR B x0x0 Example: U(x) = kx 2, the harmonic oscillator 10
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- 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
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- 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
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- x L =-Ax R =Ax0x0 You have seen this before in intro PH, but you didn't derive it this way. 13
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- 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
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- Period increases because v(x) is smaller at every x (why?) in the trajectory. Effect is magnified for larger amplitudes.
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- xLxL xRxR B x0x0 Everything is a SHO! 16
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- Equivalent angular version? 17
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- 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
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- 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
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- 20 mg L
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- Plot these to compare to SHO to pendulum 21
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- Nyquist theorem; sampling rate is critical if the sampling rate < 1/(2T), results cannot be interpreted Time (s) Displacement (degrees)