The rigid pendulum – an antique but evergreen physical model

Various kinds of motion of a rigid pendulum (including swinging with arbitrarily large amplitudes and complete revolutions) are investigated both analytically and with the help of computerized simulations. The simulation experiments reveal many interesting peculiarities of this famous physical model and complement the analytical study of the subject in a manner that is mutually reinforcing. 1. The physical system The simple pendulum is a famous physical model frequently encountered in textbooks and papers, primarily due to its important role in the history of physics. This versatile model is useful and interesting not only in itself as the most familiar example of a nonlinear mechanical oscillator, but more importantly because many problems in various branches of physics can be reduced to the differential equation describing the motion of a pendulum. The theory of solitons (solitary wave disturbances travelling in nonlinear media with dispersion), the problem of superradiation in quantum optics, and Josephson effects in weak superconductivity are the most important examples. In this paper we describe a combined analytical and computerized approach to the eternal problem of the motion of a pendulum. Our study is based on the use of an educational software package [1] developed recently by the author. The simulations allow one to investigate interesting situations which are inaccessible in a real laboratory experiment. Special attention is devoted to cases in which the swing approaches 180◦. Revolutions of the pendulum are also investigated in detail. With friction included, the differential equation for the motion of a pendulum can be written as follows: φ̈ + 2γ φ̇ + ω 0 sin φ = 0. (1) The pendulum in our model is characterized by two parameters: the angular frequency of small free oscillations ω0 = √ g/l (here l = I/ma is the reduced length, I the moment of inertia and a the axis–centre-of-mass distance), and the damping constant γ . It is convenient to use the dimensionless quality factor Q = ω0/2γ rather than the damping constant γ to measure the effect of damping. In the general case it is impossible to express the solution of the nonlinear equation (1) in terms of elementary functions, although in the absence of friction the solution can be given in terms of special functions (elliptic integrals). 0143-0807/99/060429+13$30.00 © 1999 IOP Publishing Ltd 429