A bead slides on a circular wire of radius that is being lifted vertically at a…
PHYSICS · Multi-part · Fermi University Physics · Mechanics & Dynamical Systems · Lagrangian mechanics · Lagrangian mechanics
Action and Symmetry, Ch Constraints, generalised coordinates and Lagrange's equations twice over
Problem
A bead slides on a circular wire of radius a=0.25m that is being lifted vertically at a constant speed V=0.40ms−1. Using arc length s measured along the wire from its lowest point, and writing t^ for the wire's unit tangent, someone reasons as follows: "The wire's own motion has to be included, otherwise the calculation misses the work the lift does. In a short time δt the wire rises by Vδt, so I will take the virtual displacement to be δr=t^δs+Vδtz^. Since the push N from the wire is perpendicular to t^ but not to z^, I conclude the virtual work of the push, N⋅δr, is not zero."
((a)) Find the error in this reasoning and state which definition it conflicts with.
((b)) Give the correct virtual displacement and the correct value of N⋅δr. State, separately, whether the real work done by the push over an actual time δt is zero.