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26 Nov 2019

A particle is constrained to move (without friction) on acircularwire rotating with constant angular speed ω about averticaldiameter. Find the equilibrium position of the particle,andcalculate the frequency of small oscillations aroundthisposition. Find and interpret physically a criticalangularvelocity ω=ωc that divides the particle'smotioninto two distinct types. Construct phase diagrams forthe two casesω<ωc andω>ωc.

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Physics: Principles with Applications
7th Edition, 2013
Giancoli
ISBN: 9780321625922

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Part 2 is here :

http://www.cramster.com/answers-nov-10/physics/rotating-pendulum-part-2-rotating-pendulum-part-ii-note-part-httpww_1015121.aspx

This is the same rotating pendulum of the previous two problems.Now it is time to analyze the different types of motion andequilibrium (according to the initial conditions).
Note: The “effective potential” method is a powerful tool to analyzeand solve problems where the Hamiltonian is conserved. A similartechnique is used for central potentials, including the Keplerproblem of celestial mechanics (two-body gravitationalproblem)—will see that later.
(a) Let us call Eeff ≡ H the “effective energy.” To make it look likea real energy, define Eeff = T2 + Veff , where Veff should include therelevant non-quadratic piece of kinetic energy (which effectivelybehaves, with an appropriate sign, as a potential energycontribution). In short, the effective potential is defined so thatthe effective kinetic energy left in the Hamiltonian is justquadratic.
Your task: Find the corresponding effective potential Veff (θ), i.e.,write it down explicitly as a function of θ.
Hint: Using the resolution T = T0 + T1 + T2 , it should follow thatVeff = V − T0 (using
Euler’s theorem for homogeneous functions).
(b) Analyze all possible shapes of the graph of the effectivepotential Veff (θ) (versus θ). For this purpose, simply use calculusand the techniques for energy graphs that we studied withone-dimensional systems. Specifically, you should plot Veff (θ) andfind the extrema.
Hint: There should be two different regimes (or two “families” ofshapes).
(c) At what rotation rate ωc does the stationary point at θ = 0become unstable?
(d) For ω > ωc , what is the stable equilibrium value ofθ?
Explain the physical meaning of this stable “dynamic equilibrium”value. Also explain in simple terms why θ = 0 is a stable “dynamicequilibrium” position for ω < ωc .
(e) What is the frequency Ω of small oscillations about the“dynamic equilibrium” points (for each of the two regimesdiscovered above)? Compute this frequency in two differentways:
(i) using the effective potential; and (ii) linearizing the equationof motion.

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