As it can be observed from Fig. a) and b), applying Newton s Law for the tangential force component results into: mg sin mat

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1 PHY4HF Exercse : Numercal ntegraton methods The Pendulum Startng wth small angles of oscllaton, you wll get expermental data on a smple pendulum and wll wrte a Python program to solve the equaton of moton. You wll plot the graph to vsualze the soluton (poston (angle) vs. tme). You wll also solve the energy equaton and plot t. You wll have to dscuss the output and eventually optmze the code. Background knowledge for Exercses -: Python: lsts, arrays, numercal ntegraton, scpy, pylab, leastsq. Error analyss: ch squared, goodness of the ft. R.Knght: Physcs for Scentsts and Engneers, nd ed., 8, 4.6: The Pendulum. Introducton Physcs of pendulum at small angles s based on applyng Newton s second law to derve the equaton of moton. The angle from the vertcal s θ, the dstance from the pvot pont s L; g = 9.8m/s. As t can be observed from Fg. a) and b), applyng Newton s Law for the tangental force component results nto: mg sn mat d s The tangental acceleraton s gven by: a t where the arc length dsplacement s s related to the angle θ by: s = Lθ. Brngng together all needed quanttes, we can wrte the equaton of moton of a pendulum at small angles of oscllaton as: () d g L

2 The most common method used to fndng solutons to equatons of moton s by settng up a par of coupled ordnary dfferental equatons. Gven: m = mass, q = coordnate, p = momentum and F = force, we can wrte: dp dq p F and : () m Consderng p and q to be the ntal values, we shall try to fnd solutons p( and q(.. Numercal methods To smplfy the equaton of moton, we approxmate the dervatves to: dy y( t y( t (3) Therefore, we can wrte: p( t p( F( q( ) t (4) p( q( t q( t m (4) s a set of update formulae that allow us to determne the numercal soluton (poston and momentum) at Δt, Δt, etc., gven the startng tme t = t. The numercal soluton wll approach the actual soluton as Δt. We can re-wrte (4) n a way closer to our Python code: p p F t p q q t m,,,..., t t.3 Numercal methods and the smple pendulum The equaton of moton for the smple pendulum () can be wrtten n the coupled form: d d and (6) where: Ω = g/l; θ s angle from the vertcal; ω s angular velocty. Our ntal condtons wll be: θ = 5 (small angle approxmaton) and ω =. Usng (6), the numercal approxmaton can be wrtten as: t t,,3... In equaton (7), q and s angular velocty. Ths s called the Forward Euler Method, because the rght-hand sde of (7) s evaluated at the ntal pont of the teraton step (5) (7)

3 .4 Python programmng (prelmnary) The basc steps you have to take are the followng: - defne constants - wrte ntal condtons - use numercal approxmaton (7) to step forward n tme - loop untl done - plot the graph - nterpret the result Remember that comments start wth # and the Python code s case-senstve..5 The lab exercse (All requrements marked by () have to be submtted to your TA). A pendulum consstng of a steel wre and a bob s attached to a rotatonal moton sensor, connected to a Natonal Instruments nterface. The output of the NI nterface s analyzed by a LabVew applcaton. Level the horzontal arm of the stand by usng the level provded and the knobs at the base. Measure the length of the pendulum and wegh the bob. Open the nd Yr Lab Fles folder from the lab computer desktop. Double clck on the RMS.v shortcut to open the LabVew applcaton. Before begnnng data acquston, get famlar wth cursors postons and the use of the Graph Palette to resze the graph. Clck on Acqure to start the program. In order to stop the acquston, clck on Acqure agan. The STOP button exts the program. When takng the pendulum out of equlbrum, rotate slowly the moton sensor wheel (do not touch the bob) to avod wobblng. Take the pendulum out of equlbrum by ~5 o (ths s the upper lmt of the lnear approxmaton leadng to equaton ()). Use the dsplay to setup the ntal angle. Remember that the rotatonal moton sensor sets up ts startng pont where you ntally brng the pendulum, and the LabVew applcaton reads angles n degrees. Start the acquston and let the pendulum swng for ~ seconds. In order to do the Python exercse, you wll need to determne Ω. Take the average of 5-6 oscllatons and use the cursors to obtan the perod of oscllaton T..6 Python programmng (plan) Get through ths program carefully because you may use t as a template for future applcatons. - Import the needed modules: pylab wll be needed for mathematcal lbrares and also for submodules (matplotlb.pyplo used to plottng the graphs: - Defne the needed constants: tme step Δt and Ω. - We need to calculate all the values for the plot. You took the expermental data for a total tme of seconds, so t s not a bad dea to do the calculaton up to t =. s. Gven deltat =. and t =, we ll have values. The best way to manpulate all the calculated values s to place them n an array (see the Compwk tutoral). Arrays are ndexed startng at zero. Therefore an array wth N elements has an ndex runnng from to N-. Our array wll be ntally set to zero. 3

4 - Known ntal condtons for our problem are: ntal angular velocty s zero. We have to setup the angle and the momentum ntal values. Angles have to be expressed n radans - Next step wll be wrtng the loop that ncreases the tme by Δt at each teraton. At step, the correspondng tme wll be Δt. The last tme value wll be. s - The last step wll be to plot the calculated data. We shall plot the angle on y-axs and tme on x-axs. We shall label the axes and put the plot on the screen. Wrte and run the program. Save the program on your memory stck Note: Sometmes, IDLE and Pylab do not work well together. You may need to open fewer wndows n order to keep the program stable. What happens? Are you confused? Can you nterpret the graph? To get more nformaton, try two other plots: ) angular velocty vs. tme (plot tme on x-axs and ω on y-axs) ) angular velocty vs. angle the phase plot (plot θ on x-axs and ω on y-axs). To do ths, you have to change the last lnes of code. Save each verson under a dfferent name. It could be very useful to analyze the total energy of the pendulum. All frst-year textbooks clam that energy of a smple pendulum s conserved: mv K(, U ( ) where: K s knetc energy ( K ml ) and U s E tot potental energy ( U mgh E mgl ). The energy expresson you have to use s: d ml mgl Modfy your program to calculate the energy at each step. Note that energy s not zero at t =. You have to nclude the length of the pendulum (L), the mass (m) and the gravtatonal constant (g) to the constants secton of your program. Plot: ) energy vs. tme What does the energy plot suggest? Does t explan the strange appearance of the ) and ) plots? Q. For a smple pendulum, the phase plot should be an ellpse. Usng energy conservaton, explan why. Now try to gve an explanaton for your phase plot. Q Determne the leadng error n our numercal method: perform a Taylor expanson of y(t + Δ and fnd the terms we have gnored n Equaton (4). Answer all the questons and submt all the code fles and plots to your demonstrator.. Numercal Integrators In the frst part of ths exercse, we analyzed the equaton of moton for a smple pendulum and wrote the Python code needed to numercally ntegrate the equaton. If you strctly used the numercal approxmaton, eq. (7), you must have found that everythng faled catastrophcally: angle ampltudes ncreased and energy was not conserved. The reason was the ntegraton method. We used the most prmtve numercal ntegraton method, called Euler Forward (explc: (8) 4

5 [ ] [ ] [ ] t [ ] [ ] [ ] t (9) The method ncreases the angle through an nterval Δt usng dervatve nformaton from only the begnnng of the nterval. It can be proved (see Reference at the end) that the method s unstable whch means that oscllaton ampltude as well as total energy monotoncally ncreases n tme. The numercal soluton shows a spral orbt n the (θ, ω) phase space. The method s accuracy and stablty can be mproved by decreasng the tme step, whch makes t attractve because of ts smplcty of mplementaton. Smlar to (9), the Euler Backward Method (mplc s gven by: [ ] [ ] [ ] t [ ] [ ] [ ] t The method s mplct because both [ ], [ ] are used on the rght hand sde; t s stable and therefore allows large tme steps to be taken. However, t nvolves some numercal dsspaton (see Reference). (9 ) A very smple remedy s to combne the two Euler methods nto: [ ] [ ] [ ] t () [ ] [ ] [ ] t The resultant s an explct method, stable, wth no numercal dsspaton, called Euler- Cromer or Symplectc Euler Method (SEM). Insert the new code lnes from SEM nto the program you wrote last tme and plot: Energy vs. tme and the phase plot. What happens? Compare and dscuss the phase plots from: Forward Euler, and Symplectc Method. Use the same tme step n all. Keep n mnd that n ntegratng conservatve problems t s essental to use a symplectc method.. Pendulum at large angles Wrte the Python code to ntegrate the equaton of moton of the pendulum at large angle. What you have to change are the followng: - ntal angle (n radans) - equatons of moton become: d d sn and - numercal soluton has to be re-wrtten - energy expresson s now: E ml mgl( cos ) 5

6 Use the modfed code template from wth SEM and change t to account for the large oscllaton angle. Extend the tme to mnutes. Plot Angle vs. tme, Energy vs. tme and the phase plot..3 Addng a dampng term. There s a large dscrepancy between the angle vs. tme Python plot and the real expermental data (open the RMS.v applcaton and let the pendulum swng for a couple of mnutes). It s obvous that the equatons of moton you used so far modeled the pendulum wthout dampng (physcal dsspaton). In general, the dampng force exerted on a body movng n ar or water depends on velocty v of the body relatve to the medum accordng to: C Av v () F d Coeffcents used n (3) are: C = drag coeffcent (dmensonless) A = cross-sectonal area perpendcular to the flow (m ) = densty of the medum (kg/m 3 ) v = lnear velocty of the body relatve to the medum (m/s) The drecton of the dampng (drag) force s always opposte to the drecton of velocty. The drag coeffcent C s not constant: C depends on body velocty, but also on vscosty of the medum, the shape of the body, and the roughness of ts surface. The Reynolds number R e has been found to be a useful dmensonless quantty that characterzes the dependence of the drag coeffcent on velocty. The Reynolds number s the rato of the nertal force of the medum to the vscous force: v R e () R e = Reynolds number (dmensonless) = Characterstc length of the body along the drecton of flow (m) = Dynamc vscosty of the medum (N s/m ) = Densty of the medum (kg/m 3 ) v = Lnear velocty of the body relatve to the medum (m/s) If the flow s lamnar, the Reynolds number takes small values (R e <3) and the drag coeffcent s nversely proportonal to the velocty. Ths makes the drag force drectly proportonal to the velocty: F d v When the flow s turbulent, the Reynolds number s large (4 < R e ) the drag coeffcent s approxmately constant and the drag force s dependent on the square of the velocty. Calculate the Reynolds number for the large angle pendulum. Moton and speed of a typcal pendulum bob n ar at large oscllaton angles correspond to small Reynolds numbers. Therefore, the equaton of moton of our pendulum would be wrtten as: d d sn (3) 6

7 d The dampng term was wrtten to nclude the lnear velocty: v L L. Usng cursors, take 5-6 readngs of ampltude. Assumng an exponental decay of the oscllaton envelope, estmate the decay constant γ (γ/ s the nverse of tme for whch ampltude falls to /e of the ntal value). You do not have to use Python to do ths calculaton..4 The Python applcaton The coupled equatons we need to formulate for our next Python applcaton are: d d sn (4) Use the template from part., modfy t accordng to (4) and plot Angle vs. tme. Dscuss the plot..5 Compare wth expermental data (qualtatvely) Open the RMS.v applcaton. Remember that the ntal poston of the rotatonal moton sensor sets up the orgn of the vertcal axs. Take the pendulum out of equlbrum by 3 o and let t swng for as long as you need n order to see a sgnfcant decay n ampltude. Qualtatvely compare the expermental data wth the output of your program. What do you thnk makes them dfferent? Note. Comparson wth some expermental data was justfed only for qualtatve purposes.we have not attempted to ft the data. Ths wll be the topc of the next exercse. All requrements marked by () have to be submtted to your TA. Reference: W.H. Press, S.A. Teukolsky, W.T. Vetterlng & B.P. Flannery (99), Numercal Recpes n C: the art of scentfc computng, nd ed. Cambrdge Unversty Press. Wrtten by Ruxandra Serbanescu (v.6, 9-5) 7