Homework Assignment 3 Physics 302, Classical Mechanics Fall, 2010 A. V. Kotwal Handed out: Friday, September 17, 2010 Due in class on: Friday, September 24, 2010 Problems (Ten points for each problem unless noted otherwise.) 1. Use the Lagrange multiplier method to solve the following problem: A particle in a uniform grav-itational ﬁeld is free to move without friction on a paraboloid of revolution whose symmetry axis is vertical (opening upward). Obtain the force of constraint. Prove that for a given energy and angular momentum about the symmetry axis there is a minimum and a maximum height to which the particle will go. 2. A bead is constrained to move without friction on a helix whose equation in cylindrical polar coor-dinates is ρ = b, z = aφ , with the potential V = 1 2 k ( ρ 2 + z 2 ). Use the Lagrange multiplier method to ﬁnd the constraint forces. 3. Consider a three-dimensional one-particle system whose potential energy in cylindrical polar coordi-nates ρ,θ,z is of the form
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