PHYS 217 Final, Fall 2007

Physics 217, Final Exam, January 23 2008 Time: 3 hours 1. A particle of mass m is constrained to move on the frictionless inner surface of a cone of half-angle �: Write the equations of motion of m in spherical coordinates. What are the conditions on the initial velocity v0 and initial distance r0 so that the particle moves in a circular orbit about the vertical axis. 2. An object of mass m orbits a central potential V (r) with angular momentum L. Its orbit is r = ae�b� where � is the azimuthal angle measured in the orbital plane. Find V (r): 3. Three masses m1, m2 and m3 are placed at the corners of an equilateral triangle of side s; attract each other according to Newton’s law of gravitation. a) Determine the position of the center of mass R: b) Find the resultant force F1 exerted by m2 and m3 on m1 and show that it is parallel to R: c) Show that m1 moves in a circle about the center of mass. Evaluate the radius, velocity of this orbit. d) Show that the period of this orbit is given by T = 2�s r s G (m1 + m2 + m3) 4. A perfectly smooth horizontal disk is rotating with an angular velocity ! about a vertical axis passing through its center. A person on the disk at a distance R from the center gives a perfectly smooth coin (negligible size) of mass m a push toward the origin. This push gives it an initial velocity v relative to the disk. a) Write the equations of motion for m in the x � y plane. b) Show that the equation of motion are equivalent to the single complex equation ::z + 2i! :z � !2z = 0 where z = x + iy:c) Solve this di¤erential equation by assuming that z = e�t and by taking care that in case of double root the independent solutions are of the form e�t and te�t: d) Determine the arbitrary parameters in the di¤erential equation in terms of initial distance R and velocity V: 5. A block of mass M is rigidly connected to a massless circular track of radius a on a frictionless horizontal table. A particle of mass m is con…ned to move without friction on the circular track which is vertical. a) Set up the Lagrangian using � and x as coordinates. b) Find the equations of motion. c) In the limit of small � solve the equations of motion (by eliminating ::x ) to determine � as function of time. 6. A simple pendulum of length 4l and mass m is hung from another simple pendulum of length 3l and mass m: a) Write the Lagrangian for this system and determine the equations of motion. b) Solve the system of equations for small oscillations by determining the normal modes and normal coordinates. 1