Searching for Equilibrium States of Atwood’s Machine with Two Oscillating Bodies by Means of Computer Algebra

Abstract

In this paper, we discuss the problem of searching for equilibrium states of Atwood’s machine in which a pulley of a finite radius is replaced by two separate small pulleys and both masses can oscillate in a vertical plane. Differential equations of motion for the system are derived, and their solutions are computed in the form of power series in a small parameter. It is shown that, in the case of equal masses, the equilibrium position $$r = {\text{const}}$$ of the system exists only if the oscillation amplitudes and frequencies of the masses are the same and the phase shift is α = 0 or $$\alpha = \pi $$ . In addition, there is a dynamic equilibrium state when both masses oscillate with the same amplitudes and frequencies and the phase shift is $$\alpha = \pm \pi {\text{/}}2$$ . In this case, the lengths of the pendulums also oscillate around a certain equilibrium value. Comparison of the results with the corresponding numerical solutions of the motion equations confirms their correctness. All necessary computations are carried out using the Wolfram Mathematica computer algebra system.