PHYS 339: Physical Mechanics Fall 2014 Kirkman Text: Classical Mechanics by John R. Taylor (University Science Books 2005) read chapter 1 view 20min video: 339intro.mp4 (same as: http://youtu.be/YsJadWNmwDA) problems: 1.2, 1.4, 1.19, old exam1 #1 old exam1= http://www.physics.csbsju.edu/339/2000/339t1_00.pdf http://youtu.be/oBtfxnsBoT8 same as: http://www.physics.csbsju.edu/orbit/meissel.mp4 shows three gravitationally interacting interacting "stars" with mass 3,4,5 that start (from rest) on the corners of a 3,4,5 right triangle. Note the "tails" on the stars are just to "guide the eye". Watch the video and write a few sentences that describe the motion. Please connect your words to the Newtonian words you've learned in physics. IF you can't think of much to write at least report which color star has which mass. read chapter 2 problems: 2.43, 2.44 You must have worked a problem like this in 191; now you get to work it again! Two people, one of mass 75 kg and the other of mass 55 kg sit in a rowboat of mass 80 kg. With the boat initially at rest, the two people, who have been sitting at opposite ends of of the boat 3 m apart, exchange seats. How far and in what direction will the boat move? ---above problems due 5pm Monday 1-Sept ----- read chapter 3 problem: old exam2 #1 old exam2= http://www.physics.csbsju.edu/339/2000/339t2_00.pdf This question has some pieces that are ahead of our class: Substitute these questions: (a) unchanged (b) calculate the relative velocity vector in the pre-collision and post-collision states and note that the two speeds are the same (c) Calculate the center of mass velocity in the pre-collision and post-collision states. (d) Calculate the total angular momentum in both the pre-collision and post-collision states. Calculate the orbital angular momentum in the post-collision state. Is there any spin angular momentum in the post-collision state? (e) Calculate the total kinetic energy in both the pre-collision and post-collision states. Calculate the kinetic energy "of the center of mass" in the post-collision state. 3.21, 3.22: the book says to use polar/spherical coordinates but I find these just as easy in Cartesian To find I for objects with holes: you can subtract the I of the stuff removed to make the hole. Using the subtraction idea, find the moment of inertia I of a thin spherical shell (something like a balloon: thin film--say delta r thick--surrounding nothing). Express your answer in terms of mass and radius of the shell. Using our 3d printer I made this summer a little box that is to fly on a Frisbee; see drawings top2.jpg perspective.jpg The basic box is 87x56 mm (10 mm high). The ears at the end are 7x10 mm (5 mm high) with a r=2 mm hole in the center. The inside of the box (i.e., rectangular holes) has a lip around the top edges: 2 mm deep and 2 mm from the edges. The bottom of the box is 3 mm thick (so the drop from lip to bottom is 10-3-2=5 mm) Problem: find the I of this box about its center. Note that the I of a basic rectangular plate is M (L^2+W^2)/12 (where L and W are the length and width of the sheet...the thickness does not appear in the formula). This problem will require adding and subtracting lots of pieces and using the parallel axis theorem to adjust for the parts making up the ears since their CM is not on the axis of rotation (which goes thru the box's CM). The entire mass is 17.77 g This problem is going to be a mess; I recommend setting it up in a spreadsheet. Start by finding the volume of the box; then you can find the density of the plastic (I get about .0008 g/mm^3) which will be an overall factor in your I calculation. approx answer: I=20000 g mm^2 problem: old exam3 #6 old exam3= http://www.physics.csbsju.edu/339/2000/339t3_00.pdf ---above problems due 5pm Monday 8-Sept ----- read chapter 4 problem: old exam1 #6 RE part (e)...for the given total energy the PE experienced by the particle approximates a parabola, so the motion should be approximately simple harmonic. Determine the k that would describe the corresponding SHO and figure the period for that SHO. problems: 4.2, 4.8 (see below), 4.12, 4.23 4.8.pdf gives additional background on problem 4.8 problem: old exam1 #2, #3 read chapter 5 ---above problems due 5pm Monday 15-Sept ----- 5.13 complex_review.pdf problems: 2,3,7,13,16 The file of Mathematica commands SHO1.m solves the differential equation for ten example oscillators and plots the ten resulting graphs of x vs t. The file SHO1.pdf shows these plots. Annotate each of the ten plots: directly on the hardcopy adjacent to the appropriate plot report the words that *describe* the oscillator (e.g., under damped, over damped, driven, etc) and describe how/why it *differs* from the neighboring plots. Note: describe AND compare! If you have trouble understanding/explaining a plot you can copy and paste the code into Mathematica and adjust the parameters to see what is controlling what. The file of Mathematica commands SHO2.m is similar to the above: it solves an oscillator problem that is plotted here: SHO2.pdf The results are a little puzzling (at least to me) so again describe the oscillator and describe how/why it produces this result. The Mathematica file fourier_sum.m shows the result of triangle and saw tooth driving forces (each adjusted to produce a driving force with peak-to-peak size Pi). Produce similar plots for driving forces that are half wave and pulse; adjust each to produce a driving force with peak-to-peak size Pi. Note the file fourier.m which shows how to get the FourierTrigSeries of all four of these waves. Given the results for these four waves, which produced the smallest output? Why did it produce the smallest output? old exam1 #4 greens.pdf ---above problems due 5pm Monday 22-Sept ----- exam 1: chapters 1-5 Friday 26-Sept also covered: reduced mass...see Eqs. 8.11, 8.12, 8.19 on pages 296 and 299 Help Wed 6pm on to chapter 6 HW: 6.9 6.12 ---above problems due 5pm Monday 29-Sept ----- on to chapter 7 HW: 7.14, 7.16, 7.29, 7.40; old exam3 #1 & #2 ---above problems due 5pm Monday 6-Oct ----- on to chapter 8 read: http://www.physics.csbsju.edu/orbit/orbit.2d.html Homework: Mercury_Tracking_Network_2.png shows the ground track of the 5th US manned space flight (Schirra, 3-Oct-1962). The flight was launched from FL went about 6 orbits and finished (as planned) with a splash-down in the Pacific. Note that the capsule did not fly over MN and that the orbits 'advanced' a bit (e.g. orbit 3 crossed the equator at about 10 deg and orbit 4 crossed at a bit less than 35 deg). Explain why these features were natural consequences of orbital mechanics. *Calculate* (based on orbital mechanics) the expected orbit advance and compare to the value visible in the ground track. iss_track.jpg shows some ground tracks of the International Space Station. This satellite does sometimes pass over MN. Why the difference compared to Mercury? This image may help explain: iss.gif. The altitude of ISS for these tracks was 370 km; calculate the period. (Note: the orbit is nearly circular.) IRNSSschematic.jpg shows some orbits of Indian Regional Navigational Satellite System. Explain these orbital tracks. "Super Moon" refers to a Moon that looks big because it is at perigee. What is the angular size of a super moon? What is the angular size of the Moon at apogee? (Find the orbital data for the Moon yourself.) 8.33 (this problem deals with transfer orbits) Lets plug some numbers into the above transfer orbit problem. Our book discusses "thrust ratio" but a more common figure is the change in velocity the rocket must achieve to change the orbit (delta v). Consider sending a satellite to Mars from a circular orbit like that of the Earth. Start by assuming Earth and Mars have a circular orbits with radius equal to semi major axis. (You will need to look up this and other required data on the web.) Find the two required delta v for this transfer. How long does it take to reach Mars? In fact Mars has a noticeably elliptical orbit (look up its eccentricity). Calculate the two required delta v if you join Mars when it is at perihelion. Calculate the two required delta v if you join Mars when it is at aphelion. You may assume all orbits are coplanar (of course they are not exactly so). On September 24, 2014 India's Mars probe joined Mars orbit: was Mars then closer to aphelion or perihelion? Remark: this problem is similar to: http://www.physics.csbsju.edu/364/364t199.pdf #5 8.13, 8.19, 8.23 Halley's comet has eccentricity=e=.967 semi-major axis=a=17.8 AU (FYI: 1 AU = Earth-Sun semi major axis) period=T=75.3 year What fraction of the time is the comet more than its semi major from the Sun. (Or more generally: if the eccentricity is e what fraction of the time is r>a?) http://www.physics.csbsju.edu/364/364t198.pdf #4 If you were to apply similar logic to the minor ends: the centripetal acceleration points to the empty center of the ellipse rather than the focus where the attracting mass is. What is being neglected to produce this faulty result? ---above problems due 5pm Wednesday 15-Oct ----- Note: the next chapter we cover is chapter 13 Hamiltonian Mechanics We will come back to chapter 9 following that chapter. Exam2: Friday 24-Oct chapters:6-8,13 (FYI: chapter 7 on Lagrangians will be the meat of the exam) help: 6:30 Wed 118 HAB CSB HW: hamiltonian.pdf, 13.23 ---above problems due 5pm Monday 20-Oct ----- lagrange_points.pdf: complete *one* "Option" from the Lab section For the Earth-Moon system, find the locations of the Lagrange Points: L1, L2, L3 Express your answer in terms of the Earth-Moon distance (not km) FYI: I'd use Mathematica and FindRoot; you're not going to be able to do this with algebra alone In the context of the Sun-Earth-Moon three body system (note: the big volcano crater surrounds the Sun, the little side crater is the Earth, the Moon is the satellite and the unit of distance is AU): Does the Moon orbit inside the little crater surrounding the Earth? Calculate the L1 & L2 Lagrange points for this system (Mearth/Msun=3e-6) and compare one half the distance between L1 & L2 to the Earth-Moon distance. old exam: 339t2_00.pdf #6 9.11, 9.26, 9.28, 9.30 on to chapter 10! ---above problems due 5pm Monday 3-Nov ----- old exam: http://www.physics.csbsju.edu/346/346t108.pdf #6 in chapter 10: 25,35,37,43 you might have done the below problems in 191 (i.e., just basic forces are involved) but they are well worth doing again: Find the height at which a billiard ball should be struck (horizontally) so that it will roll without slipping. Find the optimum height of the rail that bounds a billiard table. This is a details question comparing textbook Eqs. 10.93 & 10.94 to the handout "Euler Angles & Free Precession" Eqs. 13 & 14. The contexts for these equations are quite different: textbook is Euler's Equations, handout is Euler Angles, but they both deal free precession just in different ways. Compare: 13 & 10.93: do they display the same angular frequency? Compare: 14 & 10.94: they have different arrangements to show a vector rotating in the 12 plane: 14: (sin(psi), cos(psi)) 94: (cos(Omega t), -sin(Omega t)) do they both show the same direction of rotation? In the textbook omega0 is an undetermined constant from a diff eq, whereas the amplitude in the handout is a defined value...are they the same? (Hint: pphi is L, what is L*sin(theta0)? The Earth's Chandler wobble is used in the textbook as an example of free precession and as folks on Earth we are measuring in the body-fixed frame. A Google search should come up with a map showing the Chandler wobble over time: is the motion of the pole (i.e., omega) in the direction of the Earth's rotation or in the opposite direction? What direction is predicted by Eq. 14 & 10.94? From space the Earth's symmetry axis would seem to wobble. What would be the period of this motion? tetrahedron.pdf: print it out; fill in table using spreadsheet/matrix results ---above problems due 5pm Monday 10-Nov ----- Itensor.pdf coin flip: A coin in a horizontal plane is tossed into the air with angular velocity omega1 about its diameter and omega3 about its symmetry axis. IF omega3=0 the coin is flipping about its diameter; if omega1=0 the coin is simply spinning flat with the same face up. What is the smallest value of omega3/omega1 for which the wobble is such that the same face is always exposed to an overhead observer? HINT: the fact that the coin starts horizontal is an important part of the problem. HINT2: if in the intial part of the wobble the coin is horizontal, then the wobble axis cannot be vertical. FYI: you can find lots of youtube slow motion clips of coin tosses, but those I've found have omega3=0. Why is it difficult to make a pencil (spinning on point) act like a precessing gyro? Consider a typical pencil: length=15 cm; radius=.35 cm (the mass of 5 g is not needed as it will cancel). Treat the pencil as a solid cylinder (moment of tensor for a solid cylinder about its CM is given in Itensor.pdf; you need to use the parallel axis thm to figure I about an end) Using the definitions in euler_angles+top.pdf and code in euler_angles+top.m calculate "c" for the pencil-top. Run the code to find an "a" value that shows something that looks like precession with minor (but visible) nutation. Calculate the omega3 that is required for that "a" value. Provide hardcopy of the locus-of-axis (ParametricPlot3D) and the NDSolve command used. Describe in your own sentences how a boomerang works. Describe in your own sentences how a tippie-top works. tippy top video: https://www.youtube.com/watch?v=xu_Dp9IfgSU ---above problems due 5pm Monday 17-Nov ----- Note: exam3 on 24-Nov will cover chapters 9-11 (perhaps not all of 11) Help? 4:15 pm Saturday see the 339t3_00*mp4 videos for old exam solutions old exam3 #4 11.14 see lectures/Feynman_wobble.pdf ... can you find the error in the logic of the physicist called the best mind since Einstein? ---above problems due 5pm Friday 21-Nov ----- consider the following three matrices: (Note A & B are 3x3; C is 3x2) 1 2 -1 A= 0 3 1 2 0 1 2 1 0 B= 0 -1 2 1 1 3 2 1 C= 4 3 1 0 calculate: det(A), det(B), det(A.B), A.C, transpose(C).A, inverse(B), trace(A.B) report (calculation should not be needed): det(transpose(A)), det(inverse(B)), det(B.A), trace(B.A) Feel free to use your calculator or Mathematica, but (perhaps except for the inverse) these are easy to do "by hand". Find on this web site the spreadsheet diagonalizeI (It is much like the tetrahedron spreadsheet: you can input values for phi, theta, psi and the resulting Euler matrice m & transpose(m)=mT are calculated for you.) In the textbook we calculated the moment of inertia tensor for a cube using a vertex as origin: 8 -3 -3 I= -3 8 -3 -3 -3 8 (we're not bothering with the overall factor of Ma^2/12) in each of the 9 cells in box below I in the spreadsheet enter the formula that would calculate the corresponding element of the matrix m.I in the 9 cells in the box below that box and using the results for m.I you just calculated enter the formulas for the corresponding elements of m.I.mT The result is what the I matrix would look like in the new coordinate system. Enter for the Euler angles: phi=pi()/4, theta=atan(sqrt(2)), psi=0 This should diagonalize the I, i.e., we have transformed to the frame we've usually used as our "body" frame. Email me the resulting spreadsheet. (Note: if you're working within a group of 3 or fewer you can email me one copy with all the names attached. As usual, make sure all the group members understand the solution!) Note: use of the $ "hold fixed" sign on cell referecnes can save a bit of time allowing carefully constructed formulae to be "pulled down" rather than recreated from scratch. Reminder: $A1 will hold the reference to the first column fixed if the cell is copied (but the row will change from 1) A$1 will hole the reference to the first row fixed if the cell is copied (but the column will change from A) $A$1 will always refer to exactly the same cell if the cell is copied. ---above problems due 5pm Monday 8-Dec ----- Final: Monday 15-Dec 8am Alternative Final: Help: 3:15 saturday