FINAL EXAM, PHYSICS 4304, December 10, 2005

Dr. Charles W. Myles

INSTRUCTIONS: Please read ALL of these before doing anything else!!!

  1. PLEASEwrite onone sideof the paper only!! It wastes paper, but itmakes my grading easier!
  2. PLEASEdon’t write on the exam sheets, there won’t be room! If you don’t have paper, I’ll give you some.
  3. PLEASE show ALLwork, writing down the essential steps in the problem solution. Partial credit will be liberal, provided that essential work is shown. Organized work, in a logical, easy to follow order will receive more credit than disorganized work.
  4. The setup(PHYSICS)of a problem counts more heavily than the detailed math of working it out.
  5. PLEASEwrite neatly. Before handing in your solutions:PLEASE:a)Number the pages & put them in numerical order. b) Put the problem solutions in numerical order.c)Clearly mark your final answers! If I can’t read or find your answer, you can't expect me to give it the credit it deserves!d)Staple the pages together.
  6. NOTE!!“Discuss”“explain”mean to answer in terms ofPHYSICSby using a fewcomplete, grammatically correctsentences. They don’t mean to write (only) equations to answer the question!

NOTE: I HAVE 20 EXAMS TO GRADE!!! PLEASEHELP ME GRADE

THEM EFFICIENTLY BY FOLLOWING THESE SIMPLE(!!) INSTRUCTIONS!!! FAILURE TO FOLLOW THEM

MAY RESULT IN A LOWER GRADE!! THANK YOU!!

Work four (4) of the 6 problems. Each is equally weighted & worth 25 points for 100 points.

If you have read these instructions, please sign the line below, and I will add ten (10) points to your final exam grade to partially compensate for extra hours spent on evening exams and for the classes I missed during the semester. To get this credit, remember to turn this page in with your exam solutions! Note that, those who have not signed this line will clearly not have read the instructions and they will not receive this extra 10 points! Have a good Christmas Break! ______

SOME PORTIONS OF PROBLEMS 1, 2, & 3 CAN BE DONE WITHOUT THE USE OF LAGRANGE'S OR HAMILTON'S METHODS. HOWEVER, SINCE THIS IS AN EXAM WHICH IS PARTIALLY OVER THESE METHODS, NO CREDIT WILL BE GIVEN FOR SOLUTIONS TO PROBLEMS 1, 2, & 3 WHICH DON’T USE THESE METHODS!!!

  1. See Figure. The suspension point of a pendulum of length  & mass mslides without friction on a wire in the vertical x-z plane. The wire is bent into a parabola given by z = ax2(a is a constant). (Note: This is a constraint, but the constraint force isn’t needed!). Suggestion: Let (x,z) be thecoordinates of the suspension point & (X,Z) bethe coordinates of the mass. You’ll need to express (X,Z) in terms of (x,z) & the swing angle θ of the pendulum. You’ll also need to use the constraint, of course!
  1. Write expressions for the kinetic energy, the potential energy, & the Lagrangian of the system. How many degrees of freedom are there? (5 points)
  2. Derive the equations of motion using Lagrange's equations. (5 points)
  3. Derive expressions for the generalized momenta. (Caution! These are NOT simply the

mass times a velocity component!) (5 points)

  1. Write an expression for the Hamiltonian for the system. This is messy, but doable. Go as far as you can! (5 points)
  2. Derive the equations of motion using Hamilton's equations. Show that these are equivalent to the results ofb. (5 points)

NOTE!!!! Work any four (4) of the six problems!!!!

  1. The potential energy for a massmmoving in thex-y plane is V = k[x2 + y2]. It’s kinetic energy is T = m[(vx)2+ (vy)2 + a(vy)x-1 + b(vx)(vy) + c(vy)]. a,b,c,k are constants,vx, vy are the xy velocity components.
  1. Write an expression for the Lagrangianfor thissystem and derive the equations of motion using Lagrange's equations. (5 points)
  2. Derive expressions for the generalized momenta. (Caution! These are NOT simply the mass times a velocity component!). This is messy, but doable. Go as far as you can!(7 points)
  3. Write an expression for the Hamiltonian for the system. This is messy, but doable. Go as far as you can on this! (7 points)
  4. Derive the equations of motion using Hamilton's equations. (6 points)
  1. See Figure. A point mass m slides without friction down a wedge of mass M. The wedge is in the shape of an inclined plane of fixed angle α. At the same time, the wedge is sliding to the right on a frictionless, horizontal surface at constant velocity v.

a.Write expressions for the kinetic energy, the potential energy, & the Lagrangian of the

system. How many degrees of freedom are there? (5 points)

b.Derive the equations of motion using Lagrange's equations. (5 points)

c.Derive expressions for the generalized momenta of the system. (Caution! These are

NOT simply a mass times a velocity component!) (5 points)

d.Write an expression for the Hamiltonian for the system. This is messy, but doable. Go as

far as you can! (5 points)

e.Derive the equations of motion using Hamilton's equations. Show that these are

equivalent to the results of part b. (5 points)

  1. The following questions concern aparticle of massμmoving under the action of an attractive Central Force which has a potential energy of the form: U(r) =kr2.k is a positive constant. (Yes, this is an isotropic simple harmonic oscillator!)
  1. Derive an expression for the Central Force F(r). (2 points)
  2. For fixed angular momentum , write the effective potential energy V(r). SketchV(r) & qualitatively DISCUSSthe radial motion for a given particle energyE. (5 points)
  3. For a fixed particle energy E, compute the radial turning points. (Hint: This requires you to solve a quadratic equation for r2!) (5 points)

For parts d.,e. & f., assume that the particle has a known angular momentum .

  1. The is observed to be in a circular orbit of radius r0. Calculate the radius r0. (Recall: A circular orbit means that the total mechanical energy E0is equal to the minimum value of the effective potential V(r)).(5 points)
  2. Calculate the total mechanical energy E0 for the circular orbit of part d. (5 points)
  3. Calculate the speed v0 and the period T0 for the circular orbit of part d. (3 points)

NOTE:The answers to parts c, d, and e should depend on (at most) k, μ, .

NOTE!!!! Work any four (4) of the six problems!!!!

  1. Each of the following deal with a particle of mass μmoving in a central force field. Parts a and b are independent of each other and also independent of parts c and d! (HINT: The easiest wayto solve these is to use the differential equationfor the orbit, rather than the integral form.)
  1. The particle orbit r(θ) is an ellipse. That is, (α/r) = 1 + εcos(θ), where α = (2)/(μk) and ε(<1)is the eccentricity, given by ε2 = 1 + (2E2)/(μk2). Here,E (<0)is the particle energy,  is its angular momentum, and k is a positive constant. PROVE that the force F(r) has the inverse r-squared form [that is, PROVE that F(r) = -(k/r2)]. (7 points)

NOTE: By the definition of a Central Force, inparts b & c, the force function F(r)MUST depend on r ONLY, NOT & on rθ!

  1. The particle orbit is given by the equationr = 2Acosθ where A is a positive constant. Calculate the force function F(r) . (6 points)
  2. The particle orbit is given by the equation r = Aθ4 where A is a positive constant. Calculate the force function F(r). (6 points)
  3. Calculate r(t) and θ(t) for the particle orbit in part cassuming that, at t = 0, r = θ = 0. In addition to the time t, these will depend on μ, A, and. (6 points)
  1. This problem requires that you know details about elliptic orbits. In parts a –f, I want NUMBERS with proper UNITS!! An asteroid, mass m = 81020 kg, is in an elliptic orbit about the sun. The orbit eccentricity is ε = 0.65. The semi-major axis of the orbit is

a = 61011 m. (All distances are measured from the center of the sun.) The sun mass is

M = 2 1030 kg. The gravitational constant is G = 6.67 10-11 N m2/kg2. Calculate:

  1. The maximum and minimum distances of the asteroid from the sun (aphelion and perihelion), rmaxand rmin, and the semi-minor axis, b, of the elliptical orbit (4 points)
  2. The total angular momentum, , of the asteroid. (3 points)
  3. The total mechanical energy, E, of the asteroid. (3 points)
  4. The period of the orbit. (3 points)
  5. The speed, vmaxof the asteroid when it is at perihelion (rmin). Its speed, vminwhen it is at aphelion (rmax). (4 points)
  6. The speed, v of the asteroid when it is a distance r = 41011 m from the center of the sun. (4 points)
  7. Write the equation (with numerical factors explicitly evaluated) which describes the orbit r(θ). (4 points)
  1. BONUS!! There are several major differences (philosophical & calculational) between the Hamiltonian Method & the Lagrangian Method. For 5 EXTRA POINTS, in a couple of complete sentences, tell me what oneof these differences is. In our brief discussion of Ch. 8, I repeatedly emphasized these differences(in lecture, in email, & on the Web Page). If you were paying attention at all during this time, you should be able to answer this!