{{short description|Mass per unit length}} {{other uses|Density (disambiguation)}} {{Infobox physical quantity | name = Linear density | width = | background = | image = | caption = | unit = kg/m | otherunits = | symbols = <math>\rho_l</math> | dimension = wikidata | extensive = | intensive = | conserved = | transformsas = }} 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 mass density''' or simply '''linear density''' is defined in the International System of Quantities (ISQ) as the quotient of mass and length.<ref name=80000-4:2019>{{cite web|title=ISO 80000-4:2019 Quantities and units — Part 4: Mechanics|url=https://www.iso.org/standard/64975.html|publisher=International Organization for Standardization|access-date=2019-09-15}}</ref> It is also called '''titer''' in textile engineering. The SI unit of linear mass density is the kilogram per meter (kg/m).
Although (linear) density is most often used to mean (linear) mass density, the concept can be generalized for the quotient of any other quantity by length, called '''''lineic''''' quantities in the ISQ.<ref name="ISO80000-1"/> For example, ''linear charge density'' or ''lineic electric charge'' is the amount of electric charge per unit length.<ref name="ISO80000-6"/>
Linear density most often describes the characteristics of one-dimensional objects, although linear density can also be used to describe the density along one particular spatial dimension of a three-dimensional object.
==Definition== 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 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>
===Units=== Common units of linear mass density include: *kilogram per meter (using SI base units) *ounce (mass) per foot *ounce (mass) per inch *pound (mass) per yard: used in the North American railway industry for the linear density of rails *pound (mass) per foot *pound (mass) per inch
===Textile applications=== Linear density of fibers and 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 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>
Common units include: *tex, a unit of measure for the linear density of fibers, defined as the mass in grams per 1,000 meters *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 ({{xref|See also: Units of textile measurement.}})
== Generalization: lineic quantities == {{anchor|Lineic}}The qualifier '''''lineic''''' is recommended in the International System of Quantities (ISO 80000-1) to denote the quotient of any physical quantity by length.<ref name="ISO80000-1">{{cite web | title=ISO 80000-1: Quantities and units — Part 1: General | website=iso.org | url=https://www.iso.org/obp/ui/#iso:std:iso:80000:-1:ed-2:v1:en | ref={{sfnref | iso.org}} | access-date=2023-10-16}}</ref> The expressions "per unit length" or "linear ... density" (or simply "density") are also often used, with resulting units involving reciprocal metre (m<sup>−1</sup>), for example: *linear mass density or lineic mass<ref name="ISO80000-1"/><ref name=80000-4:2019>{{cite web|title=ISO 80000-4:2019 Quantities and units — Part 4: Mechanics|url=https://www.iso.org/standard/64975.html|publisher=International Organization for Standardization|access-date=2019-09-15}}</ref> *linear charge density or lineic electric charge, electric charge per unit length<ref name="ISO80000-6">{{cite web|title=IEC 80000-6:2022 Quantities and units — Part 6: Electromagnetism|url=https://www.iso.org/standard/77846.html|publisher=International Organization for Standardization|access-date=2022-11-20}}</ref> *linear number density or lineic number, number of entities per unit length *propagation constant (attenuation constant and phase constant)
===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.
== See also == * Density ** Area density ** Columnar density ** Paper density *Linear number density
== References== {{Reflist}}
{{DEFAULTSORT:Linear Density}} Category:Density Category:Length