Linear density is the measure of a quantity of any characteristic value per unit of length. Linear mass density (titer in 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.
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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
If we describe the rod as having a varying mass (one that varies as a function of position along the length of the rod,
Each infinitesimal unit of mass,
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,
The SI unit of linear mass density is the kilogram per meter (kg/m).
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 SEM 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.
Linear charge density
Consider a long, thin wire of charge
If we describe the wire as having a varying charge (one that varies as a function of position along the length of the rod,
Each infinitesimal unit of charge,
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,
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.
Units
Common units include: