A torsion pendulum has moment of inertia

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Oscillations, Waves & Acoustics · Simple, damped and driven oscillations

Conditions of Validity, Ch Linear ordinary differential equations: existence, the Wronskian, and a kernel in disguise

Problem

A torsion pendulum has moment of inertia I=2.0×10−3 kg m2I=2.0\times10^{-3}\,\mathrm{kg\,m^2}, torsional damping coefficient c=1.6×10−2 N m sc=1.6\times10^{-2}\,\mathrm{N\,m\,s} and torsional stiffness κ=50 N m rad−1\kappa=50\,\mathrm{N\,m\,rad^{-1}}, with angular displacement θ(t)\theta(t) obeying Iθ¨+cθ˙+κθ=τapp(t)I\ddot{\theta}+c\dot{\theta}+\kappa\theta=\tau_{\mathrm{app}}(t), θ(0)=0\theta(0)=0, θ˙(0)=0\dot{\theta}(0)=0. Air resistance beyond the modelled damping is ignored. At t=0t=0 a constant applied torque τ0=0.10 N m\tau_0=0.10\,\mathrm{N\,m} is switched on and held on for Δt=5.0 ms\Delta t=5.0\,\mathrm{ms}, then removed. ((a)) Write down the initial-value Green function G(t,ξ)G(t,\xi) of the operator I d2/dt2+c d/dt+κI\,\mathrm{d}^2/\mathrm{d}t^2+c\,\mathrm{d}/\mathrm{d}t+\kappa in terms of γ=c/I\gamma=c/I and ωd=κ/I−(γ/2)2\omega_d=\sqrt{\kappa/I-(\gamma/2)^2}, and verify its derivative jump at t=ξt=\xi equals 1/I1/I. ((b)) Using θ(t)=∫0tG(t,ξ)τapp(ξ) dξ\theta(t)=\int_0^tG(t,\xi)\tau_{\mathrm{app}}(\xi)\,\mathrm{d}\xi, derive a closed form for θ(t)\theta(t) valid while the torque is on, 0≤t≤Δt0\le t\le\Delta t. ((c)) Evaluate θ(Δt)\theta(\Delta t), and compare with the eventual constant-torque value τ0/κ\tau_0/\kappa.

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