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 is the physical property of matter that causes charged matter to experience a force when placed in an electromagnetic field. Electric charge can be ''positive'' or ''negative'' (commonly carried by protons and electrons respe ...
per unit length) are two common examples used in science and engineering.
The term linear 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
Function or functionality may refer to:
Computing
* Function key, a type of key on computer keyboards
* Function model, a structured representation of processes in a system
* Function object or functor or functionoid, a concept of object-oriente ...
of position along the length of the rod,
), we can write:
Each
infinitesimal 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 of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. ...
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 per
meter
The metre (British spelling) or meter (American spelling; see spelling differences) (from the French unit , from the Greek noun , "measure"), symbol m, is the primary unit of length in the International System of Units (SI), though its pref ...
(kg/m).
Linear density of
fibers and
yarns
Yarn is a long continuous length of interlocked fibres, used in sewing, crocheting, knitting, weaving, embroidery, ropemaking, and the production of textiles. Thread is a type of yarn intended for sewing by hand or machine. Modern manuf ...
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
Vibroscope ( la, vibrare 'vibrate' + scope) is an instrument for observing and tracing (and sometimes recording) vibration.
For example, a primitive mechanical vibroscope consists of a vibrating object with a pointy end which leaves a wave trace ...
. The sample is tensioned between two hard points,
mechanical vibration
Vibration is a mechanical phenomenon whereby oscillations occur about an equilibrium point. The word comes from Latin ''vibrationem'' ("shaking, brandishing"). The oscillations may be periodic, such as the motion of a pendulum—or random, such ...
is induced and the
fundamental frequency
The fundamental frequency, often referred to simply as the ''fundamental'', is defined as the lowest frequency of a periodic waveform. In music, the fundamental is the musical pitch of a note that is perceived as the lowest partial present. I ...
is measured.
Linear charge density
Consider a long, thin
wire
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 diameter 40 mm
A wire is a flexible strand of metal.
Wire is c ...
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 per
meter
The metre (British spelling) or meter (American spelling; see spelling differences) (from the French unit , from the Greek noun , "measure"), symbol m, is the primary unit of length in the International System of Units (SI), though its pref ...
(C/m).
Other applications
In
drawing 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), or density of a continuous random variable, is a function whose value at any given sample (or point) in the sample space (the set of possible values taken by the random variable) ca ...
of a single
random variable.
Units
Common units include:
*kilogram per meter
*
ounce
The ounce () is any of several different units of mass, weight or volume and is derived almost unchanged from the , an Ancient Roman unit of measurement.
The avoirdupois ounce (exactly ) is avoirdupois pound; this is the United States customa ...
(mass) per
foot
*ounce (mass) per
inch
*
pound (mass) per
yard
The yard (symbol: yd) is an English unit of length in both the British imperial and US customary systems of measurement equalling 3 feet or 36 inches. Since 1959 it has been by international agreement standardized as exactly ...
: used in the North American railway industry for the linear density of
rail
Rail or rails may refer to:
Rail transport
*Rail transport and related matters
*Rail (rail transport) or railway lines, the running surface of a railway
Arts and media Film
* ''Rails'' (film), a 1929 Italian film by Mario Camerini
* ''Rail'' ( ...
s
*pound (mass) per foot
*pound (mass) per inch
*
tex
Tex may refer to:
People and fictional characters
* Tex (nickname), a list of people and fictional characters with the nickname
* Joe Tex (1933–1982), stage name of American soul singer Joseph Arrington Jr.
Entertainment
* ''Tex'', the Italian ...
, 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
Textile fibers, threads, yarns and fabrics are measured in a multiplicity of units.
* A fiber, a single filament of natural material, such as cotton, linen or wool, or artificial material such as nylon, polyester, metal or mineral fiber, or man ...
(dtex), the
SI 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 substance's mass per unit of volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' can also be used. Mathematical ...
**
Columnar density
**
Paper density
References
{{DEFAULTSORT:Linear Density
Density