A torsion pendulum has moment of inertia and restoring torque with

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 torsion pendulum has moment of inertia I=5.0×10−4 kg m2I = 5.0\times10^{-4}\,\mathrm{kg\,m^2} and restoring torque −kθ-k\theta with k=4.5×10−3 N m rad−1k = 4.5\times10^{-3}\,\mathrm{N\,m\,rad^{-1}}, giving natural angular frequency ω0=k/I\omega_0 = \sqrt{k/I}. It is integrated by plain Euler with Δt=0.01 s\Delta t = 0.01\,\mathrm{s}. (a) Without running the integration, predict the number of steps and the corresponding number of periods after which Euler's energy has doubled. (b) State, without computing a new number, how fourth-order Runge-Kutta's energy would behave over that same number of periods, using the fact that Runge-Kutta's oscillator energy is known to drift slowly and linearly, losing only 2.7×10−32.7\times10^{-3} of its starting value after 10510^5 periods at a comparable step.

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