Editing Linear density (section)

==Linear mass density==
Consider a long, thin rod of mass <math>M</math> and length <math>L</math>.  To calculate the average linear mass density, <math>\bar\lambda_m</math>, of this one dimensional object, we can simply divide the total mass, <math>M</math>, by the total length, <math>L</math>:
<math display="block">\bar\lambda_m = \frac{M}{L}</math>
If we describe the rod as having a varying mass (one that varies as a [[Function (mathematics)|function]] of position along the length of the rod, <math>l</math>), we can write:
<math display="block">m = m(l)</math>
Each [[infinitesimal]] unit of mass, <math>dm</math>, is equal to the product of its linear mass density, <math>\lambda_m</math>, and the infinitesimal unit of length, <math>dl</math>:
<math display="block">dm = \lambda_m dl</math>
The linear mass density can then be understood as the [[derivative]] of the mass function with respect to the one dimension of the rod (the position along its length, <math>l</math>)
<math display="block">\lambda_m = \frac{dm}{dl}</math>

The [[SI]] unit of linear mass density is the [[kilogram]] per [[meter]] (kg/m).

Linear density of [[Fiber|fibers]] and [[Yarn|yarns]] can be measured by many methods. The simplest one is to measure a length of material and weigh it. However, this requires a large sample and masks the variability of linear density along the thread, and is difficult to apply if the fibers are crimped or otherwise cannot lay flat relaxed. If the density of the material is known, the fibers are measured individually and have a simple shape, a more accurate method is direct imaging of the fiber with a [[Scanning electron microscopy|scanning electron microscope]] to measure the diameter and calculation of the linear density. Finally, linear density is directly measured with a [[vibroscope]]. The sample is tensioned between two hard points, [[mechanical vibration]] is induced and the [[fundamental frequency]] is measured.<ref>{{cite journal| doi=10.1177/004051755802800809| title=Findings and Recommendations on the Use of the Vibroscope |journal=Textile Research Journal|volume=28|issue=8 |pages=691–700 |year=1958 |last1=Patt |first1=D.H.| s2cid=137534752 }}</ref><ref>{{cite web| url=http://www.iso.org/iso/iso_catalogue/catalogue_tc/catalogue_detail.htm?csnumber=6703| title=ISO 1973:1995. Textile fibres -- Determination of linear density -- Gravimetric method and vibroscope method}}</ref>