Dynamic duos: the building blocks of dimensional mechanics

Abstract

Mechanics studies the relationships between space, time, and matter, which can be expressed in terms of the dimensions of length $\mathcal{L}$, time $\mathcal{T}$, and mass $\mathcal{M}$. Each dimension broadens the scope of mechanics, from geometric quantities with dimensions of the form $\mathcal{L}^x$ (like lengths or areas), to kinematic quantities of the form $\mathcal{L}^x\mathcal{T}^y$ (like speeds or accelerations), and eventually ``mass-carrying'' quantities such as mass, force, momentum, energy, action, power, viscosity, etc. These standard mechanical quantities have dimensions of the form $\mathcal{M}\mathcal{L}^x\mathcal{T}^y$, where $x$ and $y$ are integers. In this contribution, we use this dimensional structure to arrange these mass-carrying quantities into a table indexed by $x$ and $y$. Ratios of quantities in the same rows provide characteristic lengths, and in the same columns characteristic times, encompassing a great variety of physical phenomena from atomic to astronomical scales. Most generally, we show that picking duos of mechanical quantities that are neither on the same row nor column yields dynamics, where one mechanical quantity is understood as impelling motion, while the other is impeding it. The force and the mass are the prototypes of impelling and impeding factors, but many other duos are possible. This review provides a novel synthesis revealing the power of dimensional analysis, to understand processes governed by the interplay of two mechanical quantities. This elementary decomposition of space, time and motion into pairs of mechanical factors is the foundation of ``dimensional mechanics'', a method that this review wishes to promote and advance. The review is complemented by online video lectures, which initiate a discussion on the elaborate interplay of two or more mechanical quantities.