Assignment 12 and Exam preparation

Note: we agreed that you would be allowed a 1-page "crib sheet" on the Final Exam.

Generally, this assignment is not to be handed in. It consists of problems from previous final exams in courses similar to this one (but some of the problems are rather harder than what you may expect on our Final).
However, if you want to redeem a low or non-existent previous homework grade, it may be replaced by the present homework. To decide whether it pays to do so, look up your record in the table below, eliminate the lowest grade, and then decide which homework you want to replace (note that you cannot choose 7a or 7b, only 7(a+b)). Hand in homework 12 noting which homework grade is to be replaced. The replacement will take place even if your grade on homework 12 is lower than what you want to replace.

The last column in the table ("average") is computed by the 10×sqrt(x) formula where

x = average (homework average with lowest dropped, first exam, second exam)
all on a scale of 100. Your final grade may be considerably different, because the final exam counts 40% of the total (actually between 25% and 55%, due to to the square root curve).

Problems:

  1. A spring with spring constant k is attached at one end of a mass m, located at x. The other end is located at x' and is constrained to move according to x' = A cosωt. All motion is in the horizontal x-direction only.
    1. Find the equations of motion of the mass by Lagrange's equations
    2. Show that the mass behaves as if x' were fixed and a driving force of frequency ω were exerted directly on the mass. What is the amplitude of this equivalent driving force?
    3. Write the Hamiltonian of the system. Does it equal the total energy? Is it conserved?
  2. A particle moves in a central force field characterized by the potential energy V(r) = –V0 r exp(-ar²) where V0 and a are positive constants. Discuss the particle's motion (what kind of orbits are there -- bound, unbound, circular, of all radii, are the orbits generally closed ...)
  3. A rod, length 2a and mass m turns about one end O, describing a cone with semivertical angle α. It completes a revolution in time T. Find the magnitude and direction of the angular momentm about O.
  4. A frisbee (idealized as a disk) is observed to wobble through an angle of 30° five times per second. Assume no external torques are acting on it. What is its angular speed?
  5. Text, problem 15.92
Hwk 1 Hwk 2 Hwk 3 Hwk 4 Hwk 5 1st hour exam Hwk 6 Hwk 7a Hwk7b Hwk 7(a+b) Hwk 8 Hwk 9 2nd hour exam Hwk 10 average
    5.5 4   38       0         40.5
9.5         71       0         52.7
9.3 9 9 9 10 59 9.5 2 7.5 9.5 6.5       65.5
9.5 6 5.5 4 5.5 68 8.5   4 4     30 7.5 70.6
9 5.5 7 5.5 5 54 6.5   1.5 1.5   7 50 8 73.7
9 7   7.5 6 57       0 5.5 3 60 8 75
8 6.5 3 8.5 6 64 6 1.5 5.5 7 5 3.5 50   75.3
9 8.5 2.5 9 5.5 53 1   5 5 5.5 1 70   76.3
9.8 7.5   9 5.5 62 9     0 6 5.5 60   77.5
8.5 7.5 5.5 8 8.5 61   1.5 6.5 8 8.5 7 58 9.5 79.8
9 8 6.5   6 83 8.5   7 7 6.5 4.5 55   80.8
9.5 9 7 9 5 60 8   7 7 6 7.5 60 9.5 81.5
8.7 6 4.5 8 8.5 50 9.5 2 7 9 8.5 8 78 7 83.3
10 7.5 6.5 9.5 7 73 9.5 2 6 8 7.5 7 55 8 83.5
9.7 8 6 9.5 5.5 65 9.5   6.5 6.5 6 7.5 68 7.5 83.7
9 7.5 6.5 6.5 7.5 59 7 1.5 6 7.5 7 7.5 76 9.5 83.9
9.2 5.5 7 7.5 8.5 73   2 6.5 8.5 7.5 8.5 73 6 84.8
10 8.5 8 8 6.5 88 7.5 1.5 6 7.5 7 7.5 75   88.5
10 7.5 10 9 10 73 10 1.5 5   7 8.5 80 9 88.5
8.8 9 7.5 9.5 7.5 78 8 2   2   9.5 90   88.6
9.5 9 7 9 7.5 89 9 1 7 8 6 8 70 9 90
10 9 6.5 9 6.5 85 9 1 7 8 5.5 7.5 80 5.5 90.2
8.5 7 8 7.5 7 83 7.5 1.5 6.5 8 7 7.5 85 9.5 90.7
8.5 6.5 9.5 8.5 9.5 94 10 2 7 9 9 9 98 10 97.4
10 7.5 9.5 9 9 97 9.5     9 10 9.5 95 9.5 97.7
10 10 10 10 10 96 10 2 8 10 10 8.5 100 10 99.3