
Linear density is the measure of a quantity of any characteristic value per unit of length. Linear mass density (titer in
textile engineering
Textile manufacturing or textile engineering is a major industry. It is largely based on the conversion of fibre into yarn, then yarn into fabric. These are then dyed or printed, fabricated into cloth which is then converted into useful good ...
, the amount of mass per unit length) and ''
linear charge density'' (the amount of
electric charge
Electric charge (symbol ''q'', sometimes ''Q'') is a physical property of matter that causes it to experience a force when placed in an electromagnetic field. Electric charge can be ''positive'' or ''negative''. Like charges repel each other and ...
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
and length
. To calculate the average linear mass density,
, of this one dimensional object, we can simply divide the total mass,
, by the total length,
:
If we describe the rod as having a varying mass (one that varies as a
function of position along the length of the rod,
), we can write:
Each
infinitesimal
In mathematics, an infinitesimal number is a non-zero quantity that is closer to 0 than any non-zero real number is. The word ''infinitesimal'' comes from a 17th-century Modern Latin coinage ''infinitesimus'', which originally referred to the " ...
unit of mass,
, is equal to the product of its linear mass density,
, and the infinitesimal unit of length,
:
The linear mass density can then be understood as the
derivative
In mathematics, the derivative is a fundamental tool that quantifies the sensitivity to change of a function's output with respect to its input. The derivative of a function of a single variable at a chosen input value, when it exists, is t ...
of the mass function with respect to the one dimension of the rod (the position along its length,
)
The
SI unit of linear mass density is the
kilogram
The kilogram (also spelled kilogramme) is the base unit of mass in the International System of Units (SI), equal to one thousand grams. It has the unit symbol kg. The word "kilogram" is formed from the combination of the metric prefix kilo- (m ...
per
meter
The metre (or meter in US spelling; symbol: m) is the base unit of length in the International System of Units (SI). Since 2019, the metre has been defined as the length of the path travelled by light in vacuum during a time interval of of ...
(kg/m).
Linear density of
fibers
Fiber (spelled fibre in British English; from ) is a natural or artificial substance that is significantly longer than it is wide. Fibers are often used in the manufacture of other materials. The strongest engineering materials often inco ...
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
A scanning electron microscope (SEM) is a type of electron microscope that produces images of a sample by scanning the surface with a focused beam of electrons. The electrons interact with atoms in the sample, producing various signals that ...
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
The fundamental frequency, often referred to simply as the ''fundamental'' (abbreviated as 0 or 1 ), is defined as the lowest frequency of a Periodic signal, periodic waveform. In music, the fundamental is the musical pitch (music), pitch of a n ...
is measured.
Linear charge density
Consider a long, thin
wire
file:Sample cross-section of high tension power (pylon) line.jpg, Overhead power cabling. The conductor consists of seven strands of steel (centre, high tensile strength), surrounded by four outer layers of aluminium (high conductivity). Sample d ...
of charge
and length
. To calculate the average linear charge density,
, of this one dimensional object, we can simply divide the total charge,
, by the total length,
:
If we describe the wire as having a varying charge (one that varies as a function of position along the length of the wire,
), we can write:
Each infinitesimal unit of charge,
, is equal to the product of its linear charge density,
, and the infinitesimal unit of length,
:
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,
)
Notice that these steps were exactly the same ones we took before to find
.
The
SI unit of linear charge density is the
coulomb
The coulomb (symbol: C) is the unit of electric charge in the International System of Units (SI).
It is defined to be equal to the electric charge delivered by a 1 ampere current in 1 second, with the elementary charge ''e'' as a defining c ...
per
meter
The metre (or meter in US spelling; symbol: m) is the base unit of length in the International System of Units (SI). Since 2019, the metre has been defined as the length of the path travelled by light in vacuum during a time interval of of ...
(C/m).
Other applications
In
drawing
Drawing is a Visual arts, visual art that uses an instrument to mark paper or another two-dimensional surface, or a digital representation of such. Traditionally, the instruments used to make a drawing include pencils, crayons, and ink pens, some ...
or
printing
Printing is a process for mass reproducing text and images using a master form or template. The earliest non-paper products involving printing include cylinder seals and objects such as the Cyrus Cylinder and the Cylinders of Nabonidus. The ...
, 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
In probability theory, a probability density function (PDF), density function, or density of an absolutely continuous random variable, is a Function (mathematics), function whose value at any given sample (or point) in the sample space (the s ...
of a single
random variable
A random variable (also called random quantity, aleatory variable, or stochastic variable) is a Mathematics, mathematical formalization of a quantity or object which depends on randomness, random events. The term 'random variable' in its mathema ...
.
Units
Common units include:
*kilogram per meter (using
SI base unit
The SI base units are the standard units of measurement defined by the International System of Units (SI) for the seven base quantities of what is now known as the International System of Quantities: they are notably a basic set from which al ...
s)
*
ounce (mass) per
foot
The foot (: feet) is an anatomical structure found in many vertebrates. It is the terminal portion of a limb which bears weight and allows locomotion. In many animals with feet, the foot is an organ at the terminal part of the leg made up o ...
*ounce (mass) per
inch
The inch (symbol: in or prime (symbol), ) is a Units of measurement, unit of length in the imperial units, British Imperial and the United States customary units, United States customary System of measurement, systems of measurement. It is eq ...
*
pound (mass) per
yard
The yard (symbol: yd) is an English units, English unit of length in both the British imperial units, imperial and US United States customary units, customary systems of measurement equalling 3 foot (unit), feet or 36 inches. Sinc ...
: used in the North American railway industry for the linear density of
rails
*pound (mass) per foot
*pound (mass) per inch
*
tex
Tex, TeX, TEX, may refer to:
People and fictional characters
* Tex (nickname), a list of people and fictional characters with the nickname
* Tex Earnhardt (1930–2020), U.S. businessman
* Joe Tex (1933–1982), stage name of American soul singer ...
, 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
See also
*
Density
Density (volumetric mass density or specific mass) is the ratio of a substance's mass to its volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' (or ''d'') can also be u ...
**
Area density
**
Columnar density
**
Paper density
Paper density is a paper product's mass per unit volume. The density can be calculated by dividing the grammage of paper (in grams per square metre or "gsm") by its caliper (usually in micrometres, occasionally in mils).
The "ISO 534:2011, ...
*
Linear number density
References
{{DEFAULTSORT:Linear Density
Density
Length