Remarks on Newton's Second Law for Variable Mass Systems

TL;DR

This paper clarifies misconceptions about Newton's second law for variable mass systems, demonstrating its valid forms and emphasizing its fundamental role in classical mechanics education.

Contribution

It corrects common misinterpretations by showing the validity of Newton's second law in both forms for variable mass systems and discusses its fundamental importance.

Findings

Newton's second law in the form $oldsymbol{F} = rac{doldsymbol{p}}{dt}$ is valid for variable mass systems.

The form $oldsymbol{F} = m oldsymbol{v}$ is also valid for variable mass systems.

The paper emphasizes $oldsymbol{F} = m oldsymbol{v}$ as a fundamental law in classical mechanics.

Abstract

Misinterpretations of Newton's second law for variable mass systems found in the literature are addressed. In particular, it is shown that Newton's second law in the form $\vec{F} = \dot{\vec{p}}$ is valid for variable mass systems in general, contrary to the claims by some authors that it is not applicable to such systems in general. In addition, Newton's second law in the form $\vec{F} = m \vec{v}$ -- commonly regarded as valid only for constant mass systems -- is shown to be valid also for variable mass systems. Furthermore, it is argued that $\vec{F} = m \vec{v}$ may be considered as the fundamental law, alternatively to $\vec{F} = \dot{\vec{p}}$. The present work should be of direct relevance to both instructors and students who are engaged in teaching and/or learning classical mechanics in general and variable mass systems in particular at a middle- to upper-level university physics.