Physics 106b – Problem Set 10 – Due Jan 27, 2006Version 2 – Jan 23, 2006This set covers dynamics in rotating systems and some of the early material on rigid body motion,Sections 5.1, 5.2, and 5.3.1 of the lecture notes and Chapters 7 and 8 of Hand and Finch. Be sureto have checked the lecture notes errata! Problems 1 through 4 are required, problem 5 is extracredit and equal in weight to the first four problems.Changes since v.1: Clarifications on problems 3 and 4b.1. A particle is thrown vertically upward with initial speed v0, reaches a maximum height, andfalls back to the ground. Show that the Coriolis deflection when it again reaches the groundis opposite in direction, and four time greater in magnitude, than the Coriolis deflection whenit is dropped at rest from the same maximum height.2. Prove Larmor’s Theorem: Consider a system of charged particles, all having the same ratioq/m of charge to mass, that experience mutual central forces~Fab(~rab) and are also subject toa conservative external force~Fe(~ra). Show that, if the system is subjected to a weak, uniform,constant magnetic field~B, then one can eliminate the effect of the field by observing themotion of the system in a coordinate system that is rotating relative to the initial inertialframe with angular velocity~ω = −q2 m c~B(in Gaussian units). Be sure to specify what is meant by “weak”. You may not look atGoldstein or Symon!3. In Section 5.2.3 of the lecture notes, we discuss the Lagrangian and Hamiltonian approachesto the Foucault pendulum problem. We neglected the ω2term in the kinetic energy. Repeatthe calculation of the Lagrangian and Hamiltonian with the ω2term, but setting λ =π2atthe beginning of the calculation. You should start with the equation from the notesT =12m˙x2+ ˙y2+ 2 ω sin λ (−y ˙x + x ˙y) + ω2sin2λx2+ y2+ ω2cos2λ x2Show that one obtains an equation of motion that can be derived from an effective one-dimensional potential that is the same as we would have calculated based on our study ofcentral forces. You will of course want to convert to cylindrical coordinates. Don’t forgetthe subtleties about going from the 3-dimensional to the 1-dimensional Lagrangian. Thecanonical momentum in φ picks up an extra term here, what is the significance of that term?4. Consider a uniform right circular cone of height h, half-angle α, mass M , and constant density.(a) Show that the inertia tensor of the cone in a coordinate system with the cone’s axisalong the z axis and the apex of the cone at the origin isI = M h23204 + tan2α0 003204 + tan2α00 0310tan2α1(b) T he cone rolls on its side without slipping on a uniform horizontal plane in such a mannerthat it returns to its original position in a time τ . Find expressions for the componentsof the angular velocity and angular momentum of the c one in the body and space framesand for the kinetic energy in terms of the given parameters. You may leave any inertiatensor factors in terms of the principal moments I1and I3rather than plugging in theabove messy expressions for them. Note: this problem is intended to make sure youunderstand angular velocity and can calculate angular momentum and kinetic energyof a body undergoing rigid body motion. You do not need to use Euler’s equations orcalculate the torques in the problem (though, of course you may if you would enjoy doingso.)5. In New Orleans (30◦N latitude), there was a hockey arena w ith frictionless ice. The ice wasformed by flooding a rink with water and allowing it to freeze slowly. This implies that aplumb bob would always hang in a direction perpendicular to the small patch of ice directlybeneath it. Show that a hockey puck (shot slowly enough that it stays in the rink!) will travelin a circle, making one revolution every day. (Yes, this problem is somewhat trickier than itmay appear at first glance . . .