Question

A bead of mass m is constrained to slide without friction on a circular hoop of radius R. The hoop is oriented vertically and is attached to a motor that rotates it at a constant angular speed o, as shown in the attached figure. The bead experiences a constant gravitational force directed downward, given as mg. Answer the following questions: (a) Find the Lagrangian for this system using appropriate variables. (b) Find the effective potential. (c) Find the equilibrium points for this system. For each point, state if the point is stable or unstable. Also state what, if any, requirements there are on the given quantities (m; R; g; o) for these points to exist. (d) At least one of the points you found earlier is stable. Find the period of small oscillations around the equilibrium point for one of the stable points. If there is more than one, pick the one that is easiest to compute. 0 т

please advice into missing information.

Answer 1

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We need d's -cd where co sin *2 d28 dar - (4) si (413) ( 0.4 ++) tv (04+) Une taylor expansion (cost o eq we sinten d of cos 0 (83) -> v dt For stable point we heed, w2 - sinto ey w2 to be ove. o 04 if f owl a stabe r For eq=1 unstable 2 For Coy ( 2 -> always stable. Vw2 d) Appune, f>w? ey o wh fo 2 (-4) - Mr - ) > Tz