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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 goods ...
, 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 res ...
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 M and length L. To calculate the average linear mass density, \bar\lambda_m, of this one dimensional object, we can simply divide the total mass, M, by the total length, L: \bar\lambda_m = \frac If we describe the rod as having a varying mass (one that varies as a function of position along the length of the rod, l), we can write: m = m(l) Each
infinitesimal In mathematics, an infinitesimal number is a quantity that is closer to zero than any standard real number, but that is not zero. The word ''infinitesimal'' comes from a 17th-century Modern Latin coinage ''infinitesimus'', which originally re ...
unit of mass, dm, is equal to the product of its linear mass density, \lambda_m, and the infinitesimal unit of length, dl: dm = \lambda_m dl 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, l) \lambda_m = \frac The SI unit of linear mass density is the
kilogram The kilogram (also kilogramme) is the unit of mass in the International System of Units (SI), having the unit symbol kg. It is a widely used measure in science, engineering and commerce worldwide, and is often simply called a kilo colloquially. ...
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 Fiber or fibre (from la, fibra, links=no) 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 incorporate ...
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 manu ...
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 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 co ...
of charge Q and length L. To calculate the average linear charge density, \bar\lambda_q, of this one dimensional object, we can simply divide the total charge, Q, by the total length, L: \bar\lambda_q = \frac If we describe the wire as having a varying charge (one that varies as a function of position along the length of the wire, l), we can write: q = q(l) Each infinitesimal unit of charge, dq, is equal to the product of its linear charge density, \lambda_q, and the infinitesimal unit of length, dl: dq = \lambda_q dl 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, l) \lambda_q = \frac Notice that these steps were exactly the same ones we took before to find \lambda_m = \frac. 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). In the present version of the SI it is equal to the electric charge delivered by a 1 ampere constant current in 1 second and to elementary char ...
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 Drawing is a form of visual art in which an artist uses instruments to mark paper or other two-dimensional surface. Drawing instruments include graphite pencils, pen and ink, various kinds of paints, inked brushes, colored pencils, crayo ...
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 A random variable (also called random quantity, aleatory variable, or stochastic variable) is a mathematical formalization of a quantity or object which depends on random events. It is a mapping or a function from possible outcomes (e.g., the po ...
.


Units

Common units include: *kilogram per meter * 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 a separate organ at the terminal part of the leg mad ...
*ounce (mass) per
inch Measuring tape with inches The inch (symbol: in or ″) is a unit of length in the British imperial and the United States customary systems of measurement. It is equal to yard or of a foot. Derived from the Roman uncia ("twelfth ...
* 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, 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. Mathematicall ...
** 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, ...


References

{{DEFAULTSORT:Linear Density Density