bead_on_wire - Bead moving along a thin, rigid, wire....

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Unformatted text preview: Bead moving along a thin, rigid, wire. Rodolfo R. Rosales, Department of Mathematics, Massachusetts Inst. of Technology, Cambridge, Massachusetts, MA 02139 October 17, 2004 Abstract An equation describing the motion of a bead along a rigid wire is derived. First the case with no friction is considered, and a Lagrangian formulation is used to derive the equation. Next a simple correction for the effect of friction is added to the equation. Finally, we consider the case where friction dominates over inertia, and use this to reduce the order of the system. Contents 1 Equations with no friction. 2 Principle of least action. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Equation for the motion of the bead. . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Bead on a vertical rotating hoop. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Problem 1.1: Bead on an horizontal rotating hoop. . . . . . . . . . . . . . . . . . 4 Wire on vertical plane, rotating around a vertical plane. . . . . . . . . . . . . . . . 5 Problem 1.2: Wires rotating with variable rates. . . . . . . . . . . . . . . . . . . . 5 Parametric instabilities. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 The Euler Lagrange equation: derivation. . . . . . . . . . . . . . . . . . . . . . . . . 6 Problem 1.3: Hoop moving up and down in a vertical plane. . . . . . . . . . . . . 6 Parametric stabilization. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Wire restricted to a fixed vertical plane. . . . . . . . . . . . . . . . . . . . . . . . . 7 2 Add friction to the equations. 7 Equation of motion with friction included. . . . . . . . . . . . . . . . . . . . . . . . 8 Add friction to the vertical rotating hoop. . . . . . . . . . . . . . . . . . . . . . . . 8 Wire restricted to a fixed vertical plane. . . . . . . . . . . . . . . . . . . . . . . . . 8 Wire moving rigidly up and down. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 Sliding wires and friction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1 Rosales Bead moving along a thin, rigid, wire. 2 3 Friction dominates inertia. 10 3.1 Nondimensional equations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 3.2 Limit of large friction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 4 Problem Answers. 11 1 Equations with no friction. Consider the motion of a bead of mass M moving along a thin rigid wire, under the inuence of gravity. Let ( x, y, z ) be a system of Cartesian coordinates, with z the vertical direction and z increasing with height. Let g be the acceleration of gravity. Describe the wire in parametric form, as follows: x = X ( s, t ) , y = Y ( s, t ) , and z = Z ( s, t ) , (1.1) where s is the arclength along the wire . Note that: Because the wire is thin, we have approximated it as just a curve in space....
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bead_on_wire - Bead moving along a thin, rigid, wire....

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