A pendulum of length swings through arbitrary angles (not necessarily small) and…

PHYSICS · Subjective · Fermi University Physics · Mathematical Methods for Physics · Computational Physics · Simulating classical dynamical systems

Computational Physics, Ch Stepping the equations of motion: four integrators measured

Problem

A pendulum of length LL swings through arbitrary angles (not necessarily small) and experiences a drag torque proportional to its angular velocity, so its equation of motion is θ¨=−ω02sin⁡θ−γθ˙,ω02=g/L,\ddot\theta = -\omega_0^2\sin\theta - \gamma\dot\theta,\qquad \omega_0^2 = g/L, where θ\theta is the angle from the vertical, γ\gamma is a constant damping rate, and g=9.8 m s−2g = 9.8\,\mathrm{m\,s^{-2}}. (a) Write this as a first-order system: state the two-component state vector y\mathbf{y} and the function f(y)\mathbf{f}(\mathbf{y}) such that y˙=f(y)\dot{\mathbf{y}} = \mathbf{f}(\mathbf{y}). (b) Take L=0.80 mL = 0.80\,\mathrm{m} and γ=0.15 s−1\gamma = 0.15\,\mathrm{s^{-1}}. Evaluate f\mathbf{f} numerically at the state θ=0.30 rad\theta = 0.30\,\mathrm{rad}, θ˙=0 rad s−1\dot\theta = 0\,\mathrm{rad\,s^{-1}}.

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