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Linear Density in Science & Engineering

Linear density is a measure of some characteristic value per unit length. It describes one-dimensional objects by measuring things like mass, charge, or other properties along the object's length. Common examples are linear mass density, which measures mass per length, and linear charge density, which measures charge per length. To calculate linear mass density of a rod, its total mass is divided by its total length. The linear mass density can also be understood as the derivative of the mass function with respect to the rod's length. Linear density of fibers is commonly measured by weighing a length sample, but more accurate methods involve direct imaging of fibers or using a vibroscope.

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153 views1 page

Linear Density in Science & Engineering

Linear density is a measure of some characteristic value per unit length. It describes one-dimensional objects by measuring things like mass, charge, or other properties along the object's length. Common examples are linear mass density, which measures mass per length, and linear charge density, which measures charge per length. To calculate linear mass density of a rod, its total mass is divided by its total length. The linear mass density can also be understood as the derivative of the mass function with respect to the rod's length. Linear density of fibers is commonly measured by weighing a length sample, but more accurate methods involve direct imaging of fibers or using a vibroscope.

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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, 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.
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.
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 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 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.[1][2]

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