PHYS 339: Physical Mechanics Fall 2017 Kirkman Text: Classical Mechanics by John R. Taylor (University Science Books 2005) Note: we begin with vectors and a rehash of Newtonian dynamics from 191... read chapter 1 view 20min video: 339intro.mp4 or 339intro.webm (same as: http://youtu.be/YsJadWNmwDA) Due: Monday 4 Sep problems: 1.2, 1.4, 1.10, 1.19 remark on 1.19: It is unwise to write out the equation in terms of xyz components: while it can make a proof, the algebra quickly becomes a real mess. Instead use the vectors as vectors (then the algebra is almost trivial). 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. Why does greens' exit from the screen foretell red/blue's exit? IF you can't think of much to write at least report which color star has which mass and the evidence you used to come to that conclusion. read chapter 2 problems: 2.18, 2.43 ---above due 5 pm Monday 4 Sep --- read chapter 3 problem: old exam1 #2 http://www.physics.csbsju.edu/339/handouts14/339t1_14.pdf in addition to parts a-d on the exam answer the following: e) Calculate the total angular momentum in both pre and post state...is it conserved? f) In the pre state, calculate the spin and orbital angular momentum...do they add to total L? REMARK: in text/lecture we used primes (e.g., v') to denote the relative-to-CM quantity (e.g., v'=v-V) In this test problem prime denotes the "post" situation...do not plug the v' from this problem into the formulas that involve v' from text/lecture...they mean different things! 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), starting with the I of a solid ball: (rho 4/3 pi r^3) 2/5 r^2. [The term in () is the total mass] Express your final answer in terms of mass and radius of the *shell*. Note you are basically taking the limit as delta r goes to zero, while the shell mass m remains finite. The term linear in delta r will be incorporated into the shell's mass m (m=rho 4 pi r^2 delta r), and the higher order terms will be neglected. No delta r in your final answer! FYI: could look up what the result should be in your 191 textbook Using our 3d printer I made a little box that flew on a Frisbee; see drawings top2.jpg perspective.jpg (the box held a microcomputer Flight Data Recorder [FDR] for a senior thesis on Frisbee dynamics) 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., think rectangular holes) has a lip around the top edges: 2 mm deep, 2 mm wide, 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 for rotation about the vertical axis thru its center. Note that the I of a basic rectangular plate (mass M) about a vertical axis thru its center 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. 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 http://www.physics.csbsju.edu/339/Lcatcher.pdf ---above due 5 pm Monday 11 Sep --- read chapter 4 problems: 4.2, 4.8 (see below), 4.12, 4.23 4.8.pdf gives additional background on problem 4.8 http://www.physics.csbsju.edu/339/handouts14/4.8.pdf problems: old exam1 #1, #4, #5 http://www.physics.csbsju.edu/339/handouts14/339t1_14.pdf atwood2.pdf read chapter 5 read: complex_review.pdf, do #2 & #3 at the end of the document. 5.29 Following the process of the SHO handout (which is handouts14/SHO1.m & handouts14/SHO1.pdf) plot a SHO solution of your choice. Turn in the plot, the Mathematica code, and a sentence or two that describes the problem you solved. ---above due 5 pm Monday 18 Sep --- Note: exam 1 Friday 22 Sep ---nothing due 5 pm Monday 25 Sep --- http://www.physics.csbsju.edu/339/handouts14/greens.pdf read chapter 6 on calculus of variations HW: 6.9 6.12 read chapter 7 lagrange.pdf FYI: three of the problems handed out in class were part of handouts14/339t2_14.pdf; the second exam in 339 ---above due 5 pm Monday 2 Oct --- Since Lagrange requires no consideration of typical forces of constraint, if you *want* to know the forces of constraint, more work is required. We'll talk a bit about Lagrange Multipliers (which should also be appearing in your math class). Lagrange Multipliers will not appear on exams. FYI: There are many Lagrange examples in the lectures14 folder: calculus_of_variations3+L.pdf pure Cartesian coordinates: F=ma U(r): polar coordinates, centrifugal potential, angular momentum conservation lagrange_examples.pdf sliding off sphere falling ladder atwood double atwood accelerating boxcar lagrange_examples2.pdf rolling on incline bead on a rotating circle sliding down sphere with Lagrange multipliers falling ladder with Lagrange multipliers HW 7.29 & 7.31 old exam: handouts14/339t2_14.pdf #1 b,c We'll probably be starting chapter 8 Wed After chapter 8 we'll jump ahead to chapter 13 (and then pickup at chapter 9) on to chapter 8 read: http://www.physics.csbsju.edu/orbit/orbit.2d.html HW: 8.13 Homework: In handouts14 find the following pictures: 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 heading towards Africa, 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. Why launch towards Africa? Why did it not go over MN? *Calculate* (based on orbital mechanics) the expected orbit advance and compare to the value visible in the ground track. FYI: the image cited below: iss.gif shows a 3d redering of the ISS orbit, but it may also help with Mercury. 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. (FYI: there is a wiki page for basic info.) "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.) ---above due 5 pm Monday 9 Oct --- HW 8.19 this problem is similar to: http://www.physics.csbsju.edu/364/364t199.pdf #5 so read that first. In class we calculated the transfer orbit to Mars IF mars had a circular orbit. The real Mars orbit is eccentric, so there are two extreme cases: (A) transfer to Mars when it is at perihelion (B) transfer to Mars when it is at aphelion. While our book discusses a "thrust ratio" (ratio of needed velocity to present velocity), the usual figure of interest is the change in velocity the rocket must achieve to change the orbit (delta V). To simplify, consider sending a satellite to Mars from a *circular* orbit with radius 1 AU (i.e., similar to Earth but circular). Determine the position/velocity of Mars at perihelion, and then figure the two delta V required to join Mars orbit at that perihelion from the circularized Earth orbit. (You will need to look up Mars orbit data on the web.) How long does it take to reach Mars? Determine the position/velocity of Mars at aphelion, and then figure the two delta V required to join Mars orbit at that aphelion from the circularized Earth orbit. How long does it take to reach Mars? ----- this problem explores some of the aspects of a Grand Tour starting at Jupiter. This time assume that both Earth and Jupiter have orbits that are circular (zero eccentricity), coplanar, but otherwise as in Wiki. Look up Jupiter's semi-major axis, mass, and radius. Calculate the delta V required at Earth to reach Jupiter's distance on a Hohmann transfer orbit. How long does it take? At Jupiter's location you will be moving slower than Jupiter, let's let Jupiter swat you forward. What is the approach velocity to Jupiter (V infinity). Note: we're now talking about orbital mechanics around Jupiter. What hyperbolic "a" does that correspond to? The maximum boost you can get from Jupiter will come with a perijove just above the cloud tops (i.e., the radius of Jupiter). What hyperbolic eccentricity would this extreme orbit have? In theory the best boost comes if you "bounce back" 180 degrees from Jupiter, but our eccentricity will result in a slightly smaller deflection...what deflection would be obtained? Assuming that you could arrange to have your exit velocity aligned with Jupiter's orbit velocity vector. What would this maximum exit velocity be? Is it sufficient to escape the Solar System? Using your Jupiter-relative approach velocity, at Jupiter cloud tops, you fire your engine to add 100 m/s. Your Jupiter-relative exit velocity has been increased (over the approach velocity) by how much? Finally starting directly in a 1 AU solar orbit, what delta V would be required for Solar System escape? 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?) Hint: r=a requires cos(phi)=-e (Why?)...from phi calculate the eccentric anomally u...from u calculate wt finally report the fraction of the period r>a http://www.physics.csbsju.edu/364/364t198.pdf #4 If you were to apply similar logic to the minor axes 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? HW 13.3 old exam2: handouts14/339t2_14.pdf #2 ---above due 5 pm Wednesday 18 Oct --- exam2 Friday 20 Oct help Thursday 19 Oct 7:30 pm Note: no orbital mechanics on exam2 Chapter 9 9.12b, 9.19, 9.26, 9.28 lagrange_pts.pdf (one option) old exam3: handouts14/339t3_14.pdf #3 AND (#4 OR #5) There is much interest this week in an interstellar asteroid: A/2017 U1 wiki has a page on the object and reports: Perihelion= 0.2525 ± 0.00106 AU Eccentricity= 1.1922 ± 0.00268 it says JPL has calculated the (hyperbolic) orbit and reports: Vinfinity=26 km/s "JPL solution indicates that one hundred years ago, the object was roughly 553 AU (83 billion km) from the Sun." using the wiki perihelion & eccentricity data verify the above two results. Note: hyperbolic equations summary in http://www.physics.csbsju.edu/339/handouts/hyperbola_orbit.png Note2: I get a few AU further than JPL does. ---above due 5 pm Monday 30 Oct --- chapter 10 Note: summary equations: http://www.physics.csbsju.edu/339/handouts/free_precession.png HW: in the handouts folder find and complete: Itensor.pdf and tetrahedron.pdf in chapter 10: 25,35,37 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: (-cos(psi), sin(psi)) 94: (cos(Omega t), -sin(Omega t)) do they both show the same direction of rotation? If Omega is positive, is the rotation clockwise or counterclockwise? In the textbook omega0 is an undetermined constant (oscillation amplitude) from a diff eq, whereas the amplitude in the handout is a defined value...are they the same? (Hint: ppsi is L3, what is L3*tan(theta0) in the body frame L triangle? The Earth's Chandler wobble (p 400) 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? 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". ---above due 5 pm Monday 6 Nov --- handouts : B+O_7-25.pdf Barger & Olsson (textbook authors) were professors at Madison when I was a student there. The physics building had trash cans with lids matching this problem; they indeed made fine gyroscopes. Your answer to "direction" in part a should be clockwise or counterclockwise as viewed from above. Note that the I for a cone was calculated in the notes: lectures14/moment_of_inertia2.pdf make sure your homework solution shows you understand the method---don't simply copy it! The CM of a *solid* cone was calculated in the notes: lectures14/system2_integrate.pdf Modify to find the needed CM of a hollow cone. Your numerical answer to part b should use geometry: l=15 cm; alpha=60 deg; w3 = 40 rad/s Please report the *period* for precession rather than the rate. Sketch a flying Frisbee as thrown by a right-handed person as see from above. Show: the direction of rotation, direction of L, direction of flight, and the direction of any torques (name and describe the cause of those torques). An initially straight flying Frisbee will typically hook at the end of its flight. Add an arrow to your drawing labeled "hook" showing the way the Frisbee will hook. Which way does it hook (to the right or left of the thrower)? Explain the cause of the hook. Why is it difficult to make a pencil (spinning with its point fixed) 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. Using a "b" for a simple drop initial condition, 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; unit convert to rpm. Provide hardcopy of the locus-of-axis (ParametricPlot3D) and the NDSolve command used. chapter 11: view: http://www.physics.csbsju.edu/339/solutions/7.31.webm (or if using an old browser: mp4) This video deals with problem 7.31 or in the normal mode chapter: 11.19. The video does not explain the process of finding the normal modes...thats your job in this HW problem. The video briefly shows code I've included below. w1=1 w2=2 e=1 What are these quantities in terms of the quantities mentioned in the problem (7.31 or 11.19)? m={{w2^2 -(1+e)z,-z},{-z,w1^2-z}} zroot=Solve[Det[m]==0,z] Using the Lagrangian derive the above matrix (as is done in the video). What is the relationship between the Solved values of z and the normal mode frequencies? m1=m/.First[zroot] {x10,t10}=First[NullSpace[m1]] The above {x10,t10} are then used as initial conditions to solve the differential equation: solution=NDSolve[{ (1+e) x''[t]+theta''[t] Cos[theta[t]]-theta'[t]^2 Sin[theta[t]]== -w2^2 x[t], theta''[t]+x''[t] Cos[theta[t]]== -w1^2 Sin[theta[t]] , x[0]==x10/10, x'[0]==0, theta[0]==t10/10, theta'[0]==0 },{x,theta},{t,0,50}] and it is shown that the results display a single active normal mode. Explain what the NullSpace command is finding an why the results are for a single normal mode. Do a similar calculation and provide hardcopy of a plot & code showing only the other normal mode active. handouts: ring_molecule.pdf handouts14/339t2_14.pdf #3 ---above due 5 pm Monday 13 Nov --- exam3 is Monday 20-Nov will include normal modes but not material beyond that Help? On to special relativity ---nothing due Monday 4 Dec --- homework: handouts/thomas.pdf http://www.physics.csbsju.edu/366/366t199c.pdf #6 http://www.physics.csbsju.edu/366/366t105.pdf #3, #7 use the Minkowski diagram: http://www.physics.csbsju.edu/366/minkowski.gamma=1.1.pdf http://www.physics.csbsju.edu/366/366f05.pdf #1 At exactly noon EST a boiler explodes in the basement of the Museum of Modern Art in New York City. At 0.0003 second after noon EST a similar boiler explodes in the basement of a soup factory in Camden, NJ (a distance of 150 km from the first explosion). Find a frame in which the museum explosion occurs *after* the soup factory explosion. (Report direction and speed of this frame.) Two events occur at the same place in a particular inertial frame separated by a time interval of 4 seconds. What is the spacial separation between these two events in a frame where the the two events are separated by a time interval of 6 seconds. textbook: 15.61, 15.62 ---above due Monday 11 Dec --- Final Exam: 3:30 pm Thursday 14 Dec (email if that doesn't work for you) Help?