Linear mass Density of vertically suspended heavy springs

TL;DR

This paper derives the linear mass density of heavy springs in vertical suspension scenarios by energy minimization, considering fixed and loaded conditions, providing analytical insights into their physical behavior.

Contribution

It introduces a method to calculate the linear mass density of heavy springs under different loading conditions using energy minimization techniques.

Findings

Derived formulas for linear mass density in both cases

Validated the approach through analytical solutions

Enhanced understanding of heavy spring behavior under gravity

Abstract

In the present article the linear mass Density for vertically suspended heavy springs is calculated for two different cases. First for a spring of invariable length suspended at the top and fixed at the bottom of the spring, then for a hanging heavy spring with an additional load. Both cases are solved by minimizing the total energy which is a sum of potential energy and energy due to the deformation.