MIDTERM EXAM, PHYSICS 5306, Fall, 2004

Dr. Charles W. Myles

Take Home Exam: Distributed, Monday, October 25

DUE, IN MY OFFICE OR MAILBOX, 5PM, MON., NOV. 1. NO EXCEPTIONS!

TAKE HOME EXAM RULE: You are allowed to use almost any resources (books from the

library, etc.) to solve these problems. THE EXCEPTION is that you MAY NOT COLLABORATE WITH ANY OTHER PERSON in solving them! If you have questions or difficulties with these problems, you may consult with me, but not with fellow students (whether or not they are in this class!), with other faculty, or with post-docs. You are bound by the TTU Code of Student Conduct not to violate this rule! Anyone caught violating this rule will, at a minimum, receive an “F” on this exam!

INSTRUCTIONS: Please read all of these before doing anything else!!! Failure to follow these

may lower your grade!!

  1. PLEASE write on ONE SIDEof the paper only!! This wastes paper, but it makes my grading easier!
  2. PLEASE do not write on the exam sheets, there will not be room! Use other paper!!
  3. PLEASE show all of your work, writing down at least the essential steps in the solution of a problem. Partial credit will be liberal, provided that the essential work is shown. Organized work, in a logical, easy to follow order will receive more credit than disorganized work.
  4. PLEASE put the problems in order and the pages in order within a problem before turning in this exam!
  5. PLEASE clearly mark your final answers and write neatly. If I cannot read or find your answer, you can't expect me to give it the credit it deserves and you are apt to lose credit.
  6. NOTE!!! the setup (THE PHYSICS) of a problem will count more heavily in the grading than the detailed mathematics of working it out.

PLEASE FOLLOW THESE SIMPLE DIRECTIONS!!!! THANK YOU!!!

NOTE!!!! WORK ANY 5 OUT OF THE 6 PROBLEMS! Each problem is equally weighted

and worth 20 points for a total of 100 points on this exam.

Please sign this statement and turn it in with your exam:

I have neither given nor received help on this exam ______

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  1. See the figure. Two masses, m1 and m2, are connected by a massless, inextensible string of length , which is put over the massless, frictionless pulley at the top of a wedge, as shown. The masses are allowed to slide under the influence of gravity on the two frictionless inclined planes. Use the generalized coordinates suggested in the figure to solve this problem.

a.Set up the Lagrangian for this system. How many degrees of freedom are there?

b.What is the constraint? Write the equation of constraint. What is the physical significance of the constraint force?

c.Use Lagrange’s equations along with the method of Lagrange multipliers to derive the equations of motion.

d.What are the constants of the motion? That is, what physical quantities are conserved?

e.Using the results of part c along with the equation of constraint, go as far as you can towards finding the force of constraint and solving the equations of motion.

f.Under what conditions will this system be in equilibrium? Find the equilibrium position.

NOTE!!!! WORK ANY 5 OF THE 6 PROBLEMS!

  1. See the figure. A pendulum is constructed by attaching a mass m to a massless, inextensible string of length . As shown, the upper end of the string is connected to the uppermost point on a stationaryvertical disk of radius R (R < /π). Of course, the Earth’s gravity is acting on the mass.Use the generalized coordinates suggested in the figure to solve this problem.
  1. Set up the Lagrangian for this system. How many degrees of freedom are there?
  2. Use Lagrange’s equations to find the equations of motion for this system.
  3. What are the constants of the motion?That is, what physical quantities are conserved?
  4. Using the results of part b, go as far as you can towards solving the equations of motion.
  5. Assuming small amplitudes of oscillation, obtain the oscillation frequency of this pendulum.
  1. Note: Each of the following problems deals with a particle of mass m moving in a central force field. Partsde are independent of each other and also independent of parts a, b c!
  1. The particle orbit is given by r(θ) = a(1 + cosθ) where a is a positive constant. Find the force F(r) and the corresponding potential energy V(r). NOTE! The force and the potential should be functions of rONLY(!), NOT functions of both r andθ!
  2. For the potential of part b, assume an angular momentum andfind the effective potential V´(r). Make a qualitative plot of V´(r) vs.r for different . Qualitatively discuss (using WORDS in complete, grammatically correct English sentences!) the particle orbit for different energies E. In other words, go through a discussion for this case which is similar to what Goldstein does (Sect. 3.3) in the case of the inverse r-squared force.
  3. Calculate r(t)θ(t) for the particle orbit in part a. The needed integrals are tabulated! Don’t forget the integration constants!
  4. The particle moves in a central force field given by F(r) = -(k1/r) + (k3/r3) where k1 >0 but k3can be positive or negative. Determine the conditions for a stable circular orbit of radius r0.
  5. The central force field is given by F(r) = -Kr2where K > 0. For what energy E and angular momentum  will the orbit be a circle of radius r0 about the origin? What is the period of this circular motion? If the particle is slightly disturbed from this circular motion, what will be the period of the radial oscillations about r = r0?
  1. The following problems deal with rigid body kinematics, as in Ch. 4 of Goldstein. They are from the 3rd Edition of Goldstein.
  1. Work Problem 7 (page 180).
  2. Work Problem 13 (page 181).
  3. Work Problem 16 (page 182).

NOTE!!!! WORK ANY 5 OF THE 6 PROBLEMS!!

  1. Note: The following problems concern a beam of particles of mass m scattering from a repulsive central force. Assume that the particles start at r  with velocity v0 and angular momentum .In the following, s is the impact parameter. Parts a and b are independent of partc!
  1. The central force is F(r) = (K/r4) where K > 0.Go as far as you can in calculating the scattering angle Θ(s) and the differential scattering cross section σ(Θ).
  2. The central potential is V(r) = 0, r > a; V(r) = -Kr2, r  a, with K and apositiveconstants. Go as far as you can in calculating the scattering angle Θ(s) and the differential scattering cross section σ(Θ).
  3. Work Problem 4 (page 127) of Ch. 3 of the 3rd Edition of the book by Goldstein.
  1. See the figure. A projectile of mass m, in the Northern hemisphere, at latitude λ, is launched horizontally from a height h, with initial velocity v0 towards the South.In parts ab neglectcentrifugal force effects and assume the gravitational acceleration g is the same throughout the path. For numbers, use v0 = 450 m/s, h = 100m, m = 25 kg, λ = 60º. The Earth’s angular speed isω = 7.3  10-5 rad/s. (Hint: Using a coordinate system where North isthe positive y-axis, East is the positive x-axis, and the [local] vertical is the positive z-axis maysimplify the analysis.)
  1. Ignoring the Coriolis and centrifugal forces, solve the freshman physics problem of finding the expected range R of the projectile. Put in numbers!
  2. Coriolis Effects:Compute the magnitude and direction of the Coriolis force on the projectile. Put in numbers! Compare this force to the projectile’s weight. Neglecting the centrifugal force, compute (approximately, as in class) the Coriolis deflection of the projectile and calculate the point of impact. How far South and how far West of the launch position will the projectile hit Earth? How far away from the impact point of part a will the projectile land? Put in numbers! (Hint: The Coriolis deflection due to both the vertical and the horizontal [Southward] component of the velocity will contribute. The [local] vertical and horizontal components of the Earth’s angular velocity will also both contribute to the deflection!)
  3. Centrifugal effects: Calculate the magnitude and direction of the centrifugal force on the projectile. Using an Earth radius R = 6.4  106 m, put in numbers! Compare this force to the projectile’s weight and to the Coriolis force obtained in part b. Neglecting the Coriolis force, compute (approximately, as in part b) the centrifugaldeflection of the projectile.How far to the South and how far West of the launch position will the projectile hit Earth? How far away from the impact point of part a will the projectile land? Put in numbers!
  4. When computing Coriolis effects, the centrifugal force is often neglected (except to the extent that it adds to the gravitational force to determine a “local vertical”). Based on a comparison of your results in parts b and c, in your opinion is this a reasonable approximation? Why or why not? Answer in complete, grammatically correct, English sentences!