DEPARTMENT OF PHYSICS University at Albany State University of New York Comprehensive Field Examination Classical Monday, May 21, 218 1: AM - 1: PM Instruction: Answer any four out of five questions Please check below the four problems you have done. Do not place your name on the examination booklet or this cover sheet. Each problem MUST be done in a separate examination book. The problems within each area carry equal weight. Turn in the cover sheet and the four books at the end of the exam. 1. 2. 3. 4. 5. Student Identification Code: NOTE: The same code is to be used on all sections of the Comprehensive Examination taken in May 218.
Problem 1 Three identical cylinders of mass M are arranged in a triangle as shown in the figure, with the bottom two lying on the ground. The ground and the cylinders are frictionless. You apply a constant horizontal force (directed to the right) of magnitude F on the left cylinder. For what range of F will all three cylinders remain in contact with each other and thus slide together at a constant acceleration? (Note that if you don t provide enough force the top cylinder will push the bottom two cylinders apart, while if you apply too much force the top cylinder will slip off to the left.)
Problem 2 Two particles of equal masses m move on a frictionless horizontal surface in a vicinity of a fixed force center leading to potential energies U 1 = 1 2 kr2 1 and U 2 = 1 2 kr2 2. In addition, particles interact via potential energy U 12 = 1 2 s r 1 r 2 2. Here k and s are positive constants. (a) Write the Lagrangian in terms of the center of mass position R and the relative position r = r 1 r 2. (b) Write the Hamiltonian for this system and the corresponding Hamilton s equations. (c) Solve Hamilton s equations to find R =(X, Y ) and r =(x, y) as functions of time. Page 2
Problem 3 A charged bead of mass m and electric charge q can move along a circular frictionless hoop of radius R shown in the attached picture. The hoop is centered at the origin, and it rotates with angular velocity Ω around a vertical axis. A second charge q is mounted at z = 2R, as shown in the picture. The acceleration of the gravitational field is g. (a) Write the kinetic and potential energies of the bead in terms of coordinate θ. (b) Write down the Lagrangian and derive the Euler Lagrange equations. (c) Find a transcendental equation for the equilibrium position θ of the bead (d) Identify the symmetries of the problem and write the corresponding conserved quantities, and use them to find θ(t). involving initial conditions. You may leave your answer in terms of definite integrals Page 3
Problem 4 (a) A nonconducting sphere of center O and radius R has a uniform charge density ρ. Calculate the electric field due to this sphere at all points in space. (b) A nonconducting sphere of center O and radius R has a uniform charge density ρ. nonconcentric spherical cavity of radius R <Rand of center O so that A OO = a is hollowed out of the sphere (see figure). Calculate the electric field at a point P in the cavity. (c) We now consider a conducting sphere of radius R that has no net charge. Suppose a concentric cavity of radius R <Rishollowed out of this sphere and filled with a uniformly distributed net charge +q. Calculate the electric field at all points in space. What is the total electric energy in space due to this system? Note: This problem has three independent parts. O R a O P R Page 4
Problem 5 A particle of mass m and charge q is attached to a spring of constant k, hanging from the ceiling. Its equilibrium position is a distance h above the floor. It is pulled down a distance d below equilibrium and released at time t =. Throughout the problem, assume that d λ h, whereλ is the wavelength of the radiation from the charge. (a) What is the time-averaged Poynting vector S? Note that S is a function of k, q, and the location relative to the charge (r and θ). S also depends on several constants (π, µ, c, etc.), but you can just write the product of all those constants as γ without specifying any of them. So your result should be of the form S = γ f(k, q, r, θ). (b) Calculate the intensity of the radiation hitting the floor, as a function of the distance R from the point O which lies directly below q. (c) Calculate the average energy per unit time hitting the entire floor. (d) Because the mass is losing energy in the form of radiation, the amplitude of oscillation will gradually decrease. After what time τ will this amplitude be reduced by a factor of 1/e? Hint: Start by writing the total energy of the charge as a function of amplitude. Also, assume that the fraction of energy lost in one cycle is very small, thus each cycle has a well-specified amplitude. One or more of the following integrals might be useful: (x+a) 5/2 =4/(3a 1/2 ) (x+a) 7/2 =4/(15a 3/2 ) (x+a) 9/2 =4/(35a 5/2 ) (x+a) 11/2 =4/(63a 7/2 ) Page 5