A bead of mass is threaded on a smooth

PHYSICS · Multi-part · Fermi University Physics · Mechanics & Dynamical Systems · Kinematics and Newtonian dynamics of particles · Lagrangian mechanics

Action and Symmetry, Ch Multipliers, non-holonomic constraints and forces outside the potential

Problem

A bead of mass 0.25 kg0.25\,\mathrm{kg} is threaded on a smooth, rigid, fixed circular hoop of radius a=0.60 ma = 0.60\,\mathrm{m} standing in a vertical plane, so the bead's distance from the centre is fixed, f1=r−a=0f_1 = r - a = 0. At the instant the bead passes the bottom of the hoop its speed is 3.0 m s−13.0\,\mathrm{m\,s^{-1}}. Air resistance is ignored. Take g=9.8 m s−2g=9.8\,\mathrm{m\,s^{-2}} and add the constraint to the Lagrangian as L+λ1f1L+\lambda_1f_1. ((a)) Keeping rr as a coordinate and f1=r−af_1 = r - a as the constraint, find the value of the multiplier λ1\lambda_1 at this instant. ((b)) State the magnitude and direction of the force the hoop exerts on the bead there.

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