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Time scale for the rapid transient (overdamped bead on a rotating hoop).

The governing equation m*r*φ'' = -b*φ' - m*g*sinφ + m*r*(ω^2)*sinφ*cosφ, can be reduced to ε(d^2φ/dτ^2) + dφ/dτ = f(φ).

Using phase plane analysis it can be shown that the equation ε(d^2φ/dτ^2) + dφ/dτ = f(φ) has solutions that rapidly relax to the curve where dφ/dτ = f(φ). Here f(φ) = sinφ(γcosφ - 1), ε = (m^2*g*r)/b^2, τ = t/T, and T is chosen to be T = b/(m*g).

a) Estimate the time scale T_fast for this rapid transient in terms of ε, and then express T_fast in terms of the original dimensional quantities m, g, r, ω, and b.

b) Rescale the original differential equation, using T_fast as the characteristic time scale, instead of T_slow = b/mg. Which terms in the equation are negligible on this time scale?

c) Show that T_fast ≪ T_slow if ε≪1. (In this sense, the time scales T_fast and T_slow are widely separated.)

Relevant equations

γ = (r*ω^2)/g

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# Time scale for the rapid transient

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