Mass as a form of Energy in a simple example

TL;DR

This paper illustrates how mass and energy are equivalent by analyzing a simple system of masses and springs, making the relativistic concept more accessible through familiar classical examples.

Contribution

It provides a clear, non-relativistic example demonstrating how potential energy contributes to the total mass of a bound system, bridging classical mechanics and relativity.

Findings

Potential energy adds to the system's mass in a simple mass-spring model.

The non-relativistic limit clarifies the mass-energy relationship.

The example aids in understanding Einstein's mass-energy equivalence.

Abstract

A major consequence of special relativity, expressed in the relation $E_0 = m c^2$, is that the total energy content of an object at rest, including its thermal motion and binding energy among its constituents, is a measure of its inertia, i.e. its mass. This relation was first stated by Einstein. He showed that, in order to be consistent with the principles of special relativity, there must be a loss of inertia in a block that emits two pulses of electromagnetic radiation. A pedagogical difficulty with this example is that radiation is a purely relativistic phenomenon, and so the connection with the examples one learns in introductory Mechanics courses is not simple. Here we use a more familiar example of masses and springs, where the non-relativistic limit can be easily found and where the potential energy is clearly shown to be part of the mass of the bound system.