Name: PHYS2330 Intermediate Mechanics Fall 2010 Final Exam Tuesday, 21 Dec 2010 This exam has two parts. Part I has 20 multiple choice questions, worth two points each. Part II consists of six relatively short problems, worth ten points each. The short problems can be worked out on the front page of the sheet provided, but use the back if you need more room. In any case please be neat! Also, two extra pages are provided at the back. Use these if you need to, but be sure to indicate which problem you are working on these pages. You may use your textbook, course notes, or any other reference you may have other than another human. You are welcome to use your calculator or computer, although the test is designed so that these are not necessary. Good luck! Part I (Total): Part II Problem 1: Problem 2: Problem 3: Problem 4: Problem 5: Problem 6: Total:
Part I: Multiple Choice Questions (Two points each) 1. An object with mass m moves vertically near the Earth s surface. If y measures the vertical coordinate, with y increasing moving upward, the Hamiltonian is A. 1 2 my2 + mgy B. 1 2 my2 mgy C. 1 2 mẏ2 + mgy D. 1 2 mẏ2 mgy E. 1 2 mẏ2 + mgẏ 2. Which of the following fields can represent a conservative force? A. F(x, y) =xyˆx + xŷ B. F(x, y) =yˆx xyŷ C. F(x, y) =xy(ˆx + ŷ) D. F(x, y) =x 2ˆx + y 2 ŷ E. F(x, y) =yˆx 3. A force in one dimension is given by F (x) =x 2. The potential energy function is A. 2x B. 2x C. x 2 D. x 3 /3 E. x 3 /3 4. A 2 kg mass is attached to a spring with stiffness constant k = 50 N/m. The mass is also subject to a linear drag force, with drag coefficient b = 16 N/(m/sec). At what (angular) frequency does this system oscillate? A. 2/sec B. 3/sec C. 4/sec D. 5/sec E. 6/sec
5. In a certain coordinate system, the matrix representation of a stress tensor in a continuous medium has Σ xy = Σ yx = σ while all other elements are zero. The force on an area element da lying in the yz plane is A. zero. B. σdaˆx C. σdaŷ D. σdaẑ E. σda(ŷ + ẑ)/ 2 6. The following diagram plots time dependent responses x(t) for a nonlinear system, as a function of some kind of driver r: For which values of r do you expect x(t) to eventually be completely different for only slight changes in the initial conditions? A. r 3.45 B. r 3.62 C. 3.59 r 3.61 D. 3.81 r 3.82 E. r =3.45, 3.55, 3.57,...
7. A particle of mass m moves in a plane, subject to a force F = 3kr 2 where r is the distance from the origin. If φ measures the particle s angular position with respect to the x-axis, the Lagrangian is A. mṙ 2 /2+m φ 2 /2 kr 3 B. mṙ 2 /2+m φ 2 /2+kr 3 C. mṙ 2 /2+mr 2 φ2 /2 kr 3 D. mṙ 2 /2+mr 2 φ2 /2+kr 3 E. mṙ 2 /2+mr 2 φ2 /2 3kr 2 8. An asteroid is in a circular orbit about the Sun. The radius of the orbit is about nine times the distance of the Earth to the Sun. How long does it take the asteroid to complete one orbit? A. Three years. B. Nine years. C. Eighteen years. D. Twenty-seven years. E. Eighty-one years. 9. A pendulum bob hangs from a string in a laboratory located precisely at the Earth s equator. The bob is not moving. It hangs A. directly towards the center of the Earth. B. nearly towards the center of the Earth, but slightly north. C. nearly towards the center of the Earth, but slightly south. D. nearly towards the center of the Earth, but slightly east. E. nearly towards the center of the Earth, but slightly west. 10. An object of mass m moves according to the Hamiltonian H(x, p) =Ap 2 + Bx 4 where A and B are constants. The Hamiltonian equations of motion are A. ẋ =2Ap and ṗ = Bx 4 B. ẋ = p/m and ṗ = Bx 4 C. ẋ =2Ap and ṗ = Bx 4 D. ẋ = p/m and ṗ = 4Bx 3 E. ẋ =2Ap and ṗ = 4Bx 3
11. An inviscid, incompressible fluid flows horizontally through a pipe of constant cross sectional area with a uniform velocity of 10 m/s. If the fluid flows from left to right, the difference in pressure between two points 100 m apart is closest to which of the following? (Assume the density of the fluid is 1000 kg/m 3.) A. zero. B. 10 Pa C. 10 3 Pa D. 10 5 Pa E. 10 7 Pa 12. A rigid body rotates with angular velocity ω = ωˆx in a certain (x, y, z) coordinate system. The moment-of-inertia tensor, as a matrix in the same coordinate system, is The angular momentum vector L is A. mr 2 ωˆx B. mr 2 ω(2ŷ +6ẑ) C. mr 2 ω(ˆx +2ŷ +6ẑ) D. mr 2 ω(ˆx +3ŷ +5ẑ) E. mr 2 ω(5ˆx +4ŷ +6ẑ) I = mr 2 1 3 5 3 2 4 5 4 6 13. To a very good approximation, the speed of light is one foot per nanosecond. In a certain reference frame, event A occurs 4 feet away and 5 nanoseconds before event B. In another reference frame, events A and B occur simultaneously. How far apart, spatially, are they separated in the second reference frame? A. 3 feet. B. 4 feet. C. 5 feet. D. They happen at the same place. E. In no reference frame can A and B occur simultaneously.
14. A simple pendulum is made from a bob of mass m hanging from a massless rod of length. It swings in a plane and θ measures the angle from the vertical. If θ never gets very large, the potential energy function U(θ) is A. mgθ B. (g/)θ 2 C. g 2 θ 2 /2 D. mgθ 2 /2 E. mg sin θ 15. Three equal masses m are located at the corners of an equilateral triangle of side length a. The moment of inertia for rotations about an axis parallel to the plane of the triangle, and passing through two of the corners, is A. ma 2 /2 B. 3ma 2 /4 C. ma 2 D. 2ma 2 E. 3ma 2 16. Which of the following is not a characteristic of a system in chaotic motion? A. Positive Liapunov exponent. B. Nonlinear equation of motion. C. Extreme sensitivity to initial conditions. D. Fractal behavior in a bifurcation diagram. E. A lack of periodic motion in response to a periodic driving term. 17. A small planet interacts with a massive star through their mutual gravitational attraction. Which of the following is not a possible shape of its orbit? A. a circle. B. an ellipse. C. a parabola. D. a hyperbola. E. a straight line.
18. Which of the following functions satisfies the wave equation 2 f/ x 2 = 2 f/ t 2? A. f(x, t) =x 2 t 2 B. f(x, t) =x 2 + t 3 C. f(x, t) = sin(xt) D. f(x, t) =cos(x 2 t 2 ) E. f(x, t) = ln[(x t) 3 /(x + t) 2 ] 19. A uniform circular disk of mass M and radius R rolls without slipping on a horizontal surface with speed v. Its kinetic energy is A. Mv 2 /2 B. 3Mv 2 /4 C. Mv 2 D. 5Mv 2 /4 E. 3Mv 2 /2 20. Which of the following names is not associated with a key contribution to the development of classical mechanics? A. Euler B. Hamilton C. Lagrange D. Napolitano E. Newton
Part II: Short Problems (10 points each) 0.65 Problem 1. The plot shows the effective potential for a planet orbiting a massive star, as a function of the distance of the planet from the star, in certain units. A planet in this potential takes one year to execute a circular orbit. For an elliptical orbit of a planet with the same mass, but with total energy E = 1, find the eccentricity and the orbital period. Effective potential 0.7 0.75 0.8 0.85 0.9 0.95 1 1.05 1.1 2 3 4 5 6 7 8 Distance from star
Problem 2. A particle of mass m moves in one dimension x, and is acted on by a force F (t) =F 0 e bt. The particle starts from rest at t = 0 with initial position x = 0. Find its position x(t) as a function of time.
Problem 3. Apply the analysis concepts we used for the damped driven pendulum to analyze a (under-)damped driven linear oscillator, namely the differential equation φ +2β φ + ω 2 0φ = γω 2 0 cos ωt In particular, (a) sketch a bifurcation diagram, (b) find the Liapunov exponent, (c) sketch the state space orbit, and (d) draw a Poincaré section. It is not necessary to put in values for the various parameters, but you are welcome to do so if you like. Clearly label the axes of all plots, however. I am looking for just the general behavior, but any detail you would like to provide can help me understand if things don t look quite right.
Problem 4. A simple plane pendulum consists of a massless rod of length and a bob of mass m. Let φ measure the displacement from vertical. The upper end of the rod is attached to a point which accelerates horizontally with constant acceleration a. Construct the Lagrangian function L(φ, φ, t) and determine the differential equation of motion for φ. Show that the equation of motion linearizes to that of a simple harmonic oscillator for small φ. Find the frequency and equilibrium angle φ 0 for the small oscillations.
Problem 5. A long, thin cylindrical rod has mass M, length L and cross sectional radius R. Give values for (all three) principle moments of inertia. (You are welcome to derive them, but you may also quote them from other sources.) At time t = 0 the rod has angular velocity ω 0 at an angle θ with respect to the rod, find the angular velocity ω 3 (t) along the long axis of the rod as a function of time.
Problem 6. The circulation of a fluid with velocity field v(x,t) is defined as Γ C = C v d for some closed loop C. Consider the steady flow of an incompressible inviscid fluid which has zero circulation everywhere, with gravity being the only active volume force. Show that 1 2 ρv2 + ρgz + p =0 (Note that this is the same quantity we studied in Bernoulli s theorem, which stated that this is a constant in time. You are asked here to show that it is constant in space.) You may use without proof the vector calculus theorem v ( v) = 1 2 v2 (v )v.
Extra Paper #1: Please show the problem on which you are working!
Extra Paper #2: Please show the problem on which you are working!