A bead of mass is threaded on a smooth wire shaped
PHYSICS · Multi-part · Fermi University Physics · Mechanics & Dynamical Systems · Oscillations and normal modes · Lagrangian mechanics
Action and Symmetry, Ch Constraints, generalised coordinates and Lagrange's equations twice over
Problem
A bead of mass m=0.015kg is threaded on a smooth wire shaped, near its lowest point, so that its height z above that point is related to arc length s measured from it by z(s)≈s2/(2R), with radius of curvature R=0.35m. A film of oil on the wire exerts on the bead a drag force −bs˙t^ tangential to the wire, opposing its sliding relative to the wire, with b=0.0060kgs−1. The wire is bolted to the floor of a lift car, so every point of the wire shares the lift's vertical acceleration Z¨(t). Take g=9.8ms−2, and assume the bead's oscillation stays small enough that z(s)≈s2/(2R) holds throughout.
((a)) With the lift at rest, use d'Alembert's principle to obtain the bead's small-oscillation equation of motion, and find the period of the resulting damped oscillation.
((b)) The lift now accelerates upward at a steady Z¨=1.5ms−2. Find the new period.
((c)) Find the time for the oscillation's amplitude to fall to 1/e of its value, and state whether this time changes when the lift accelerates.
((d)) In one sentence, say what component the bead's real displacement over a short time interval has, while the lift is accelerating, that no virtual displacement of the bead ever has.