A bead slides without friction on a circular hoop of radius that spins at a constant…

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Mechanics & Dynamical Systems · Nonlinear dynamics and chaos

Modes, Packets and Media, Ch The anharmonic oscillator: frequency, harmonics and drift

Problem

A bead slides without friction on a circular hoop of radius R=0.100 mR=0.100\,\mathrm{m} that spins at a constant angular speed Ω\Omega about its vertical diameter. With θ\theta the bead's angle from the bottom of the hoop, measured along the hoop, its equation of motion is θ¨=−gRsin⁡θ+Ω2sin⁡θcos⁡θ,\ddot\theta=-\frac gR\sin\theta+\Omega^2\sin\theta\cos\theta, with g=9.81 m s−2g=9.81\,\mathrm{m\,s^{-2}}. ((a)) Show that for Ω2>g/R\Omega^2>g/R the bead has an equilibrium off the axis at cos⁡θ∗=g/(RΩ2)\cos\theta^*=g/(R\Omega^2), and find the Ω\Omega that puts it at θ∗=60∘\theta^*=60^\circ. ((b)) Write θ=θ∗+x\theta=\theta^*+x and expand the equation of motion to third order in xx. Show that it takes the standard form with ω02=Ω2sin⁡2θ∗\omega_0^2=\Omega^2\sin^2\theta^*, η=32Ω2sin⁡θ∗cos⁡θ∗\eta=\tfrac32\Omega^2\sin\theta^*\cos\theta^* and ε=16Ω2(7cos⁡2θ∗−4)\varepsilon=\tfrac16\Omega^2(7\cos^2\theta^*-4), and evaluate all three for θ∗=60∘\theta^*=60^\circ. ((c)) Show that the bead softens for every off-axis equilibrium, and find the fractional frequency shift at θ∗=60∘\theta^*=60^\circ for a swing of amplitude A=0.100 radA=0.100\,\mathrm{rad}. ((d)) Find the mean-position drift at that amplitude, and say whether the bead's average position moves towards the bottom of the hoop or away from it.

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