EXAM II, PHYSICS 4304, April 20, 2009
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 is no 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 a 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. Problems for which just answers are shown, without the work being shown, will receive ZERO credit!
4. The setup (PHYSICS) of a problem counts more than the mathematics of working it out. In problems requiring integration, remember that the lower limit on an integral might be important!
5. PLEASE write neatly. Before handing in the solutions, PLEASE: a) put problem solutions in numerical order, b) number the pages & put them in order, & c) clearly mark your 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 write English sentences in the answer. They DON’T mean to answer using only symbols. Answers to such questions containing only symbols without explanation of what they mean will get ZERO CREDIT!!! It would also be nice if upper level physics students would try to write complete, grammatically correct English sentences!
NOTE: I HAVE 8 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!!
NOTE: You MUST work Problem 1 (on the Lagrangian method). Work ANY THRE (3) of the remaining 4 problems for a total of 4 problems required. Each is equally weighted & worth 25 points, for a total of 100 points possible.
Please NOTE also the 10 point bonus questions below!!!
BONUS QUALITATIVE QUESTIONS!! Worth 2 points each for a total of 10 Bonus Points on this Exam!! Answer parts a, b, c and d briefly in English, keeping mathematical symbols to a minimum!
a. Define the concept of Deterministic Chaos. EXPLAIN the difference between Deterministic Chaos and Randomness.
b. Using mostly English words, state Newton’s Universal Law of Gravitation. If you insist on writing a formula, DEFINE EVERY SYMBOL you use!
c. Using mostly English words, state Gauss’s Law for Gravitation. If you insist on writing a formula, DEFINE EVERY SYMBOL you use!
d. Using mostly English words, state Hamilton’s Principle. If you insist on writing a formula, DEFINE EVERY SYMBOL you use!
e. In class, more than once I outlined what I called the “recipe” for using the Lagrangian Method to obtain the equations of motion in a mechanical system. List the basic steps in this “recipe”. DEFINE EVERY MATHEMATICAL SYMBOL you use!
NOTE!!!! PROBLEM 1 is required!!
1. NOTE!! Obviously, the following problem can be done without the use of Lagrange’s method. But, since this exam is partially over this method, no credit will be given if you don’t use this method!!! See figure. Consider the plane pendulum as shown. It is of a point mass m is attached to one end of a massless, inextensible rod of length ℓ. The other end of the rod is attached to a frictionless hinge on a horizontal ceiling. The hinge allows movement only in the plane (the x-y plane) of the figure (characterized by angle θ). The only external force acting is gravity.
a. Using Cartesian (rectangular) coordinates (x, y) and the corresponding velocity components, write expressions for the potential energy, the kinetic energy, & the Lagrangian for this system. How many degrees of freedom are there? (Hint: Are the x and y for the mass independent?)
b. Use Lagrange’s equations to derive the equations of motion for x and y.
c. Starting with the results of parts a and b, transform the problem to plane polar coordinates (r, θ). Write expressions for the potential energy, the kinetic energy, & the Lagrangian for this system using those coordinates. Is there a constraint? If so, what is it?
d. Use Lagrange’s equations to derive the equation of motion for θ.
NOTE!!!! Work any three (3) of problems 2, 3, 4, & 5!!
HINT for Problems 2 & 3!
Recall that, for situations of high symmetry, field calculations are often much easier if Gauss’s Law is used!
2. The following questions are about a solid, infinitely long right circular cylinder of radius R which has a non-uniform volume mass density, which depends on the distance r from the cylinder’s axis as ρ = ρ0(r2/R2), where ρ0 is a constant density. Please read the HINT, just before this problem, before starting it! Calculate:
a. The gravitational field g outside the cylinder at a distance r R from the axis. (7 points)
b. The gravitational field g inside the cylinder at a distance r R from the axis. (8 points)
c. The gravitational potential at all points in space. Take the zero of potential at r = R. (10 points)
The following facts about a cylinder of radius r & length ℓ might be useful:
Lateral surface area: A = 2πrℓ. Volume: V = πr2ℓ.
This integral might be useful: ∫rn dr = (rn+1)/(n+1); n is any power (n ¹ -1).
Don’t forget the lower limit!!!
NOTE!!!! Work any three (3) of problems 2, 3, 4, & 5!!
3. The following questions are about a sphere of radius R which has uniform volume mass density ρ. Please read the HINT, just before Problem 2, before starting this problem! Calculate:
a. The gravitational field g outside the sphere & a distance r R from the center. (7 points)
b. The gravitational field g inside the sphere and a distance r R from the center. (8 points)
c. The gravitational potential at all points in space. Let the zero of potential be at r = ¥. (10 points)
The following facts about a sphere of radius r might be useful:
Surface area: A = 4πr2. Volume: V = (4πr3).
This integral might be useful: ∫rn dr = (rn+1)/(n+1); n is any power (n ¹ -1).
Don’t forget the lower limit!!!
NOTE!!!! Work any three (3) of problems 2, 3, 4, & 5!!
4. NOTE: Parts a., b., c. & d. are (obviously!) independent of each other! Work each with Newton’s Universal Gravitation Law (or results obtained from it in Ch. 5) NOT by assuming a constant gravitational acceleration! All parts concern a fictional planet X. Assume that X is a sphere with uniform mass density. Data to obtain the required numerical results: Gravitation constant: G = 6.67 ´ 10–11 N·m2/kg2, Planet Mass: MX = 8.0 ´ 1024 kg, Planet Radius: RX = 5.5 ´ 106 m.
a. A particle of mass m is shot vertically from X’s surface at a speed v0 = 8.5 ´ 103 m/s. Calculate the maximum height which it will reach before it begins to fall back to the surface. A numerical result is wanted & needed! (4 points)
b. A particle of mass m is dropped from rest at a height h = 7.5 ´ 105 m above X’s surface. Calculate the speed with which it will hit the surface. A numerical result is wanted & needed! (3 points)
c. Calculate the escape velocity for a particle of mass m on the surface of X. (That is, find the minimum velocity needed to “escape” from X’s gravity field). A numerical result is wanted & needed! (4 points)
d. A spherical satellite of uniform density & mass m = 2,500 kg is in a circular orbit of radius
r = 3.0 ´ 107 m about X. Calculate the gravitational force between the satellite & X, the
speed of the satellite in orbit, & the period of the satellite’s orbit. Calculate the effective
gravitational acceleration (in m/s2) experienced by the satellite. Numerical results are
wanted & needed! (4 points)
e. A particle of mass m is dropped from rest from a large height h above X’s surface. By using conservation of total mechanical energy, derive an expression for the time t it takes to fall to X’s surface. This will be in the form of a messy integral. Leave it in integral form! (10 points)
NOTE!!!! Work any three (3) of problems 2, 3, 4, & 5!!
5. NOTE: Parts a. and b. are (obviously!) independent of each other!
a. By a successive approximation procedure, use your calculator to solve the equation
x - 1 = 2sin(x). Obtain a result accurate to 4 significant figures. Use either direct iteration or Newton’s Method. I recommend the latter! Hint: Your initial “guess” should be some x > 0. Before choosing this initial guess, it might be useful to make a sketch of x - 1 and 2sin(x) on the same graph to give you an idea where the two functions cross. (13 points)
b. Using complete, grammatically correct English sentences, define the following terms which concern chaos: limit cycle, Poincaré section, mapping, bifurcation, Lyapunov exponents. (12 points)