Departent of Physics Preliinary Exa January 3 6, 2006 Day 1: Classical Mechanics Tuesday, January 3, 2006 9:00 a.. 12:00 p.. Instructions: 1. Write the answer to each question on a separate sheet of paper. If ore than one sheet is required, staple all the pages corresponding to a single question together in the correct order. But, do not staple all probles together. This exa has five questions. 2. Be sure to write your exa identification nuber (not your nae or student ID nuber!) and the proble nuber on each proble sheet. 3. The tie allowed for this exa is three hours. All questions carry the sae aount of credit. Manage your tie carefully. 4. If a question has ore than one part, it ay not always be necessary to successfully coplete one part in order to do the other parts. 5. The exa will be evaluated, in part, by such things as the clarity and organization of your responses. It is a good idea to use short written explanatory stateents between the lines of a derivation, for exaple. Be sure to substantiate any answer by calculations or arguents as appropriate. Be concise, explicit, and coplete. 6. The use of electronic calculators is perissible and ay be needed for soe probles. However, obtaining preprograed inforation fro prograable calculators or using any other reference aterial is strictly prohibited. The Oklahoa State University Policies and Procedures on Acadeic Dishonesty and Acadeic Misconduct will be followed.
Attept all five probles. Each proble carries 20 points. Proble 1 A disk is ounted on a shaft through the center of ass of a block which slides down a frictionless plane incline at an angle θ as shown in the figure. The radius of the disk is a and the axle is offset a distance b fro the center of the disk. The ass of the disk is and its oent of inertia about an axis through the center of ass is I. The ass of the block is M. A b φ C θ θ The axle A passes through the center of ass of the block of ass M. The disk has a ass. The disk is pivoted on an axle through A at a distance b fro the center of the disk C. I is the oent of inertia about C. The generalized coordinates are X and φ; φ is easured fro the line through A which is perpendicular to the plane. θ is the angle ade by the incline with the horizontal. See the figure and note the definition of the angle φ which is easured fro the line (plane) perpendicular to the incline. (a) Write down the Lagrangian for the syste in ters of the generalized coordinates X (the center of ass of the block) and φ.
(b) Show that the equations of otion are (M + )Ẍ + b φ cos φ b φ 2 sin φ (M + )g sin θ =0 and bẍ cos φ + b2 φ + I φ + gb sin(φ θ) =0 (c) Show that φ(t) 0 (for all tie) is a solution. Why does the disk hang at an angle with respect to the vertical? Interpret this result in ters of the non-inertial frae defined by the sliding block. (d) Show that the equation of otion for the disk can be written as (b 2 + I) φ 2 b 2 ( φ cos 2 φ φ 2 sin φ cos φ) (M + ) +(gb cos θ) sin φ =0 (e) Find the solution to part (d) for the case where M and the aplitude of the otion is sall, φ 1. (f) In the special case where θ = 0 (and again with M ) and initial conditions φ = φ 0, φ = 0, show that once the disk is released, the disk swings and the block rocks.
Proble 2 The European Space Agency is now accepting European contractors bids for the Don Quijote space ission that will test the collision technique for altering asteroid orbits. The ission ight be launched as early as 2011. This test is part of a larger effort to develop effective deflection ethods for protecting Earth fro the ipacts of large ( > 1 k) asteroids. v M ω (a) The target asteroid will have a diaeter of about 500, and a ean density near 2000 kg/ 3. Use those nubers to copute the asteroid s ass M, assuing that it is spherical. (b) The interceptor spacecraft will have a ass of 400 kg, and hit the asteroid with a relative speed v of 15 k/s. A copanion spacecraft will easure how the ipact affects the asteroid s velocity. Assuing that the collision is perfectly inelastic, copute the agnitude of the asteroid s velocity change v in the liit M. (c) The copanion spacecraft will additionally easure how the ipact affects the asteroid s rotation. As a first step in evaluating rotational effects, show that the oent of inertia I of a uniforly dense sphere of ass M and radius R rotating about any diaeter is I = 2 5 MR2 (d) Suppose that the target asteroid is spherical, uniforly dense, and spinning with a typical rotational period of 6 hours on an axis through its center. Copute its angular frequency ω in rad/s.
(e) Calculate the agnitude of the asteroid s angular frequency change ω of the asteroid after ipact in the liit M. Assue that the asteroid s oent of inertia reains unchanged, and that the collision occurs tangentially, in the equatorial plane of the asteroid, and in its direction of rotation. Proble 3 Three point particles of equal ass are connected by three assless springs obeying Hooke s law and having spring constants 4k, 7k and 11k respectively. All three springs have equal unstressed lengths l. The spring with spring constant 4k is attached to the ceiling, as shown in the figure. The syste is disturbed so that the asses oscillate in a vertical straight line. 4 k 7 k 11 k (a) Deterine the equilibriu configuration of the syste. That is, deterine the positions of the three asses in equilibriu. (b) Construct the Lagrangian for the syste. (c) Obtain the noral odes and the noral frequencies for sall oscillation in the vertical direction. You ay leave the noralization of the noral odes arbitrary. (Hint: Look for factoring of the eigenvalue equation.) (d) Sketch qualitatively the relative aplitudes of the noral odes.
Proble 4 The equation for the radial otion of a ass in a central potential is r = r V (r)+ l2 2r 2 r U(r) where U(r) =V (r)+ and the constant of otion l is defined by l2 2r 2 r 2 φ = l (a) Show that a second constant of otion is E = 1 2 ṙ2 + U(r) (b) Assue V (r) = 1 2 kr2. Then, show that, for a given l, the radius of the circular orbit, the energy, and the angular velocity are given by r 0 =(l 2 /k) 1/4, E =(kl 2 /) 1/2, φ = l r 2 0 =(k/) 1/2 ω. (c) Further assue that this circular orbit is suddenly perturbed by a radial ipulse. Deterine (approxiately) the perturbed orbit, showing that r(t) r 0 + A cos 2ωt and φ(t) ωt A r 0 sin 2ωt where A r 0 is the aplitude of the radial displaceent fro the circular orbit.
Proble 5 A unifor bar of ass M and length 2l is suspended fro one end by a assless spring of force constant k and equilibriu length d. The bar can swing freely only in one vertical plane, and the spring is constrained to ove only in the vertical direction. (a) Set up the Lagrangian for the syste. (b) Find the Hailtonian for the syste and obtain the Hailton s equations of otion. (c) Are there any ignorable coordinates? What quantities are conserved in the proble?