Editing Linear density

{{Short description|Measure of a quantity of any characteristic value per length}}
{{other uses|Density (disambiguation)}}
[[File:Linear density along a rod.svg|thumb|317x317px|The linear density, represented by λ, indicates the amount of a quantity, indicated by m, per unit length along a single dimension.]]
'''Linear density''' is the measure of a quantity of any characteristic value per unit of length.  '''Linear mass density''' ('''titer''' in [[Textile Engineering|textile engineering]], the amount of mass per unit length) and ''[[linear charge density]]'' (the amount of [[electric charge]] per unit length) are two common examples used in science and engineering.

The term linear density or linear mass density is most often used when describing the characteristics of one-dimensional objects, although linear density can also be used to describe the density of a three-dimensional quantity along one particular dimension.  Just as density is most often used to mean mass density, the term linear density likewise often refers to linear mass density.  However, this is only one example of a linear density, as any quantity can be measured in terms of its value along one dimension.

==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>

==Linear charge density==
{{main|Linear charge density}}

Consider a long, thin [[wire]] of charge <math>Q</math> and length <math>L</math>.  To calculate the average linear charge density, <math>\bar\lambda_q</math>, of this one dimensional object, we can simply divide the total charge, <math>Q</math>, by the total length, <math>L</math>:
<math display="block">\bar\lambda_q = \frac{Q}{L}</math>
If we describe the wire as having a varying charge (one that varies as a function of position along the length of the wire, <math>l</math>), we can write:
<math display="block">q = q(l)</math>
Each infinitesimal unit of charge, <math>dq</math>, is equal to the product of its linear charge density, <math>\lambda_q</math>, and the infinitesimal unit of length, <math>dl</math>:<ref>{{Citation
 | last1 = Griffiths
 | first1 = David J.
 | title = Introduction to Electrodynamics (2nd Edition)
 | place = New Jersey
 | publisher = [[Prentice Hall]]
 | pages = [https://archive.org/details/introductiontoel00grif/page/64 64]
 | year = 1989
 | isbn = 0-13-481367-7
 | url-access = registration
 | url = https://archive.org/details/introductiontoel00grif/page/64
 }}</ref>
<math display="block">dq = \lambda_q dl</math>
The linear charge density can then be understood as the derivative of the charge function with respect to the one dimension of the wire (the position along its length, <math>l</math>)
<math display="block">\lambda_q = \frac{dq}{dl}</math>

Notice that these steps were exactly the same ones we took before to find <math display="inline">\lambda_m = \frac{dm}{dl}</math>.

The [[SI]] unit of linear charge density is the [[coulomb]] per [[meter]] (C/m).

==Other applications==

In [[drawing]] or [[printing]], the term linear density also refers to how densely or heavily a line is drawn.

The most famous abstraction of linear density is the [[probability density function]] of a single [[random variable]].

==Units==
{{See also|Units of textile measurement}}
Common units include:
*kilogram per meter (using [[SI base unit]]s)
*[[ounce]] (mass) per [[Foot (unit)|foot]]
*ounce (mass) per [[inch]]
*[[Pound (mass)|pound]] (mass) per [[yard]]: used in the North American railway industry for the linear density of [[Rail tracks|rail]]s
*pound (mass) per foot
*pound (mass) per inch
*[[Units of textile measurement#Tex|tex]], a unit of measure for the linear density of fibers, defined as the mass in grams per 1,000 meters
*[[Units of textile measurement#Denier|denier]], a unit of measure for the linear density of fibers, defined as the mass in grams per 9,000 meters
*[[decitex]] (dtex), a unit for the linear density of fibers, defined as the mass in grams per 10,000 meters

== See also ==
* [[Density]]
** [[Area density]]
** [[Columnar density]]
** [[Paper density]]
*[[Linear number density]]

== References==
{{Reflist}}

{{DEFAULTSORT:Linear Density}}
[[Category:Density]]
[[Category:Length]]