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FINAL EXAM, PHYSICS 4304, December 12, 2007
Dr. Charles W. Myles
INSTRUCTIONS: Please read ALL of these before doing anything else!!!
1. PLEASE write on one side of the paper only!! It wastes paper, but it makes my grading easier!
2. PLEASE don’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 ALL work, writing down at least 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 mathematics of
working it out.
5. PLEASE write neatly. Before handing in your solutions, PLEASE: a) number the pages &
put them in numerical order, b) put the problem solutions in numerical order, and 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.
6. NOTE!! The words “Discuss” & “explain” mean to answer in terms of PHYSICS by using
a few complete, grammatically correct English sentences. They don’t mean to write (only)
equations! The word “Sketch” means to make at least a rough drawing of the item you are asked to
sketch.
NOTE: I HAVE 10 EXAMS TO GRADE!!! PLEASE HELP 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 5 problems. Each is equally weighted & worth 25 points for 100 points
If you have read these instructions, please sign the line below, & I will add five (5) points to your final exam grade to
partially compensate for extra hours spent on evening exams. 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!
____________________________________
NOTE: 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!!!
NOTE!!!! Work any four (4) of the five problems!!!!
1. See figure. Two masses, m1 & m2, are connected by a massless,
inextensible string of length ℓ, which is put over the massless, frictionless
pulley at the top of a wedge, as shown. m1 & m2, are allowed to slide under
the influence of gravity on the 2 frictionless inclined planes. As suggested
in the figure, use the generalized coordinates (ℓ1,θ1,ℓ2,θ2) to solve this.
a. Write expressions for the kinetic energy, the potential energy, & the Lagrangian. How many
degrees of freedom are there? (5 points)
b. What is the constraint? Write the equation of constraint. What is the physical significance of the
constraint force? (4 points)
c. Use Lagrange’s method to derive the equations of motion. (Use the form without Lagrange
multipliers. First, explicitly use the constraint in the Lagrangian of part a.) (5 points)
d. Derive expressions for the generalized momenta & write an expression for the Hamiltonian. This
is messy, but doable. Go as far as you can! (6 points)
e. Derive the equations of motion using Hamilton's equations. Show that these are equivalent to the
results of c. (5 points)
NOTE!!!! Work any four (4) of the five problems!!!!
2. See figure. A pendulum is constructed by attaching a mass m to a massless, inextensible string of
length ℓ. As shown, the upper end of the string is connected to the uppermost point on a stationary
vertical disk of radius R (R < ℓ/π). At any time, a portion of the string of
s
length u isn’t touching the disk & a portion of length s is touching it.
Obviously, u & s are time dependent. Of course, Earth’s gravity acts on the
mass. As suggested in the figure, use the generalized coordinates (u,s,θ)
suggested in the figure to solve this problem.
a. Write expressions for the kinetic energy, the potential energy, & the
Lagrangian. How many degrees of freedom are there? (5 points)
b. What is the constraint? Write the equation of constraint. What is the
u
physical significance of the constraint force? (4 points)
c. Use Lagrange’s equations to find the equations of motion for this system.
(Use the form without Lagrange multipliers. First, explicitly use the constraint in the Lagrangian of
part a.) (5 points)
d. Derive expressions for the generalized momenta & write an expression for the Hamiltonian. This
is
messy, but doable. Go as far as you can! (6 points)
e. Derive the equations of motion using Hamilton's equations. Show that these are equivalent to the
results of c. (5 points)
3. In Cartesian coordinates, the potential energy for a mass m moving in the xy plane is V = k[x2 + y2].
It’s kinetic energy is T = m{(vx)2 + (vy)2 + a[(vx)y + (vy)x]}. a & k are constants, vx,vy are the x & y
velocity components.
a. Write an expression for the Lagrangian for this system and derive the equations of motion using
Lagrange's equations. (6 points)
b. Derive expressions for the generalized momenta. (Caution! These are NOT simply the mass times
a velocity component!). (6 points)
c. Write an expression for the Hamiltonian for the system. This is messy, but doable. Go as far as you
can on this! (7 points)
d. Derive the equations of motion using Hamilton's equations. (6 points)
4. Note: Each of the following problems deals with a particle of mass μ moving in a central force field.
Parts d is independent of parts a, b & c!
a. The particle orbit is given by r(θ) = a(1 + cosθ) where a is a positive constant. Find the force F(r)
& the corresponding potential energy U(r). NOTE! F(r) & U(r) should be functions of r
ONLY(!),
NOT functions of both r & θ! (6 points) (HINT: The easiest way to solve for F(r) is to use the
differential equation for the orbit, rather than the integral form.)
b. For the potential of part a, assume an angular momentum ℓ & find the effective potential V(r).
Make a qualitative SKETCH of V(r) vs. r for different ℓ. Qualitatively DISCUSS (using WORDS
in
complete, grammatically correct English sentences!) the particle orbit for different energies E. (6
points)
c. Calculate r(t) & θ(t) for the particle orbit in part a. See below for the needed integrals! Don’t
forget
the integration constants! (7 points)
d. The central force field is given by F(r) = -Kr2 where K > 0. For what energy E and angular
momentum ℓ will the orbit be a circle of radius r0 about the origin? What is the period of this
circular
motion? (6 points)
The following integrals might be useful. Constants of integration are not shown:
∫dθ cos(θ) = sin(θ)
∫dθ cos2(θ) = (½)θ + (¼)sin(2θ)
NOTE!!!! Work any four (4) of the five problems!!!!
5. This problem requires that you know some details about elliptic orbits. I want NUMBERS with
proper UNITS!! Halley’s Comet, mass m = 5 1015 kg, is in an elliptic orbit about the sun. This orbit
has very high eccentricity ε = 0.97. The semi-major axis of the orbit is a = 2.7  1012 m (measured
from the center of the sun). Sun mass M = 2 1030 kg. Gravitation constant G = 6.67 10-11 N m2/kg2.
Calculate:
a. The maximum & minimum distances of the comet from the sun (aphelion & perihelion), rmax &
rmin, & the semi-minor axis, b, of the orbit (5 points)
b. The total angular momentum ℓ, of the comet. (4 points)
c. The total mechanical energy, E, of the comet. (4 points)
d. The period of the orbit. (3 points)
e. The speed, vmax of the comet when it is at perihelion (rmin). Its speed, vmin when it is at aphelion
(rmax). (Hint: You can use a conservation law to help you find these.) (4 points)
f. The speed, v of the comet when it is a distance r = 5 1011 m from the center of the sun. (5
points)
6. BONUS!! There are several major differences (philosophical & calculational) between the
Hamiltonian Method & the Lagrangian Method of obtaining the equations of motion for a system. For
5 EXTRA POINTS, in a couple of complete sentences, tell me what one of these differences is. In
our discussion of Ch. 7, I repeatedly emphasized these differences (in lecture & on the Web Page). If
you were paying attention at all during this time, you should be able to answer this!