Dumbbell dynamics: a didactical approach

TL;DR

This paper introduces a simplified model using a mechanical dumbbell to explain tidal dissipation effects, illustrating phenomena like lunar orbit changes and Earth's rotational slowdown suitable for educational purposes.

Contribution

It presents an accessible, scale-analysed model to demonstrate tidal dissipation effects, including orbital and rotational evolution, aimed at high school and college students.

Findings

Tidal dissipation causes lunar orbit circularization and enlargement.

Earth's rotation slows down due to tidal effects.

Tidal dissipation influences spin-orbit resonance regimes.

Abstract

In this paper we propose a simplified model to describe the dissipative effects of tides. We assume a spherical Earth with a dissipative coupling with a mechanical dumbbell. The latter has a mass much smaller than the Earth's, and it models the presence of the tidal bulges. Using properly the scale analysis, we will show that some of the consequences of tidal dissipation are the circularization and the enlargement of orbit of the Moon and the slowing down of the Earth's rotation. We will also see that tidal dissipation plays a fundamental role for the establishment of a regime of spin-orbit resonance in the celestial systems. The mathematical tools used make our treatment appropriate for senior high school students or college students.