Classical Mechanics (Kibble & Berkshire) · Problem 5.10

Energy conservation — T.W.B. Kibble, F.H. Berkshire

Physics — Mechanics · driven-linear-oscillations

Problem

A driven oscillator satisfies the equation x¨+Ω2x=F0cos⁡[Ω(1+ϵ)t],\ddot{x} + \Omega^2 x = F_0\cos[\Omega(1+\epsilon)t], where ϵ\epsilon is a positive constant. Show that the solution that satisfies the initial conditions x=0x = 0 and x˙=0\dot{x} = 0 when t=0t = 0 is x=F0ϵ(1+12ϵ)Ω2sin⁡12ϵ Ωt sin⁡Ω(1+12ϵ)t.x = \frac{F_0}{\epsilon(1+\frac{1}{2}\epsilon)\Omega^2}\sin\tfrac{1}{2}\epsilon\,\Omega t\,\sin\Omega(1+\tfrac{1}{2}\epsilon)t. Sketch the graph of this solution for the case in which ϵ\epsilon is small.

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