A block of mass on a spring of stiffness starts at rest at its natural length, ,

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 block of mass m=0.50 kgm = 0.50\,\mathrm{kg} on a spring of stiffness k=200 N m−1k = 200\,\mathrm{N\,m^{-1}} starts at rest at its natural length, x(0)=0x(0)=0, x˙(0)=0\dot{x}(0)=0, and obeys mx¨+kx=F(t)m\ddot{x}+kx=F(t), where FF is the applied force and xx the displacement from equilibrium. Friction and air resistance are ignored, and the spring is ideal (Hookean for all displacements considered). ((a)) Show that the initial-value Green function of the operator m d2/dt2+km\,\mathrm{d}^2/\mathrm{d}t^2+k is G(t,ξ)=sin⁡(ω0(t−ξ))mω0(t>ξ),G(t,ξ)=0(t<ξ),G(t,\xi)=\frac{\sin(\omega_0(t-\xi))}{m\omega_0}\quad (t\gt\xi),\qquad G(t,\xi)=0\quad (t\lt\xi), with ω0=k/m\omega_0=\sqrt{k/m}, and that x(t)=∫0tG(t,ξ)F(ξ) dξx(t)=\int_0^t G(t,\xi)F(\xi)\,\mathrm{d}\xi agrees with the variation-of-parameters formula built from cos⁡ω0t\cos\omega_0t and sin⁡ω0t\sin\omega_0t. ((b)) A constant force F(t)=F0=10 NF(t)=F_0=10\,\mathrm{N} is switched on at t=0t=0 and held on thereafter. Carry out the convolution integral to show that x(t)=(F0/k)(1−cos⁡ω0t)x(t)=(F_0/k)(1-\cos\omega_0t). ((c)) Evaluate x(t)x(t) at t=0.10 st=0.10\,\mathrm{s}.

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