An accurate formula for the period of a simple pendulum oscillating beyond the small-angle regime

A simple approximation formula is derived here for the dependence of the period of a simple pendulum on amplitude that only requires a pocket calculator and furnishes an error of less than 0.25% with respect to the exact period. It is shown that this formula describes the increase of the pendulum period with amplitude better than other simple formulas found in literature. A good agreement with experimental data for a low air-resistance pendulum is also verified and it suggests, together with the current availability/precision of timers and detectors, that the proposed formula is useful for extending the pendulum experiment beyond the usual small-angle oscillations.Comment: 15 pages and 4 figures. to appear in American Journal of Physic

Critical wave-packet dynamics in the power-law bond disordered Anderson Model

We investigate the wave-packet dynamics of the power-law bond disordered one-dimensional Anderson model with hopping amplitudes decreasing as $H_{nm}\propto |n-m|^{-\alpha}$. We consider the critical case ($\alpha=1$). Using an exact diagonalization scheme on finite chains, we compute the participation moments of all stationary energy eigenstates as well as the spreading of an initially localized wave-packet. The eigenstates multifractality is characterized by the set of fractal dimensions of the participation moments. The wave-packet shows a diffusive-like spread developing a power-law tail and achieves a stationary non-uniform profile after reflecting at the chain boundaries. As a consequence, the time-dependent participation moments exhibit two distinct scaling regimes. We formulate a finite-size scaling hypothesis for the participation moments relating their scaling exponents to the ones governing the return probability and wave-function power-law decays