Vibrations and Waves · Problem 4-5

Coupled oscillators and normal modes — A.P. French

Physics — Waves · forced-oscillations-pendulum

Problem

A simple pendulum has a length (ll) of 1 m. In free vibration the amplitude of its swings falls off by a factor ee in 50 swings. The pendulum is set into forced vibration by moving its point of suspension horizontally in SHM with an amplitude of 1 mm. (a) Show that if the horizontal displacement of the pendulum bob is xx, and the horizontal displacement of the support is ξ\xi, the equation of motion of the bob for small oscillations is d2xdt2+γdxdt+gl x=gl ξ\frac{d^2x}{dt^2} + \gamma \frac{dx}{dt} + \frac{g}{l}\,x = \frac{g}{l}\,\xi Solve this equation for steady-state motion, if ξ=ξ0cos⁡ωt\xi = \xi_0 \cos \omega t. (Put ω02=g/l\omega_0^2 = g/l.) (b) At exact resonance, what is the amplitude of the motion of the pendulum bob? (First, use the given information to find QQ.) (c) At what angular frequencies is the amplitude half of its resonant value?

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