A bead of mass slides without friction on a rigid circular hoop of radius fixed in a…

PHYSICS · Multi-part · Fermi University Physics · Mechanics & Dynamical Systems · Lagrangian mechanics · Lagrangian mechanics

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

Problem

A bead of mass

m=0.15\,\mathrm{kg}

slides without friction on a rigid circular hoop of radius

a=0.40\,\mathrm m

fixed in a vertical plane. At the bottom of the hoop the bead's speed is

3.0\,\mathrm{m\,s^{-1}}

(

g=9.8\,\mathrm{m\,s^{-2}}

). Someone writes the constraint as

f_1=r^2-a^2=0

, forms

L=\tfrac12m(\dot r^2+r^2\dot\theta^2)-mgr\cos\theta+\lambda_1(r^2-a^2)

(with

\theta

measured from the top), works through the radial Lagrange equation at

r=a

, and finds

\lambda_1=-6.06\,\mathrm{N\,m^{-1}}.

The solution then concludes: "so the hoop pushes the bead with a force of

6.06\,\mathrm N

." ((a)) Find the error in this conclusion. ((b)) Give the correct value of the push, in newtons.