Linear density

Although (linear) density is most often used to mean (linear) mass density, the concept can be generalized for the quotient of any other quantity by length, called lineic quantities in the ISQ. For example, linear charge density or lineic electric charge is the amount of electric charge per unit length.

Linear density most often describes the characteristics of one-dimensional objects, although linear density can also be used to describe the density along one particular spatial dimension of a three-dimensional object.

Definition

Consider a long, thin rod of mass ${\displaystyle M}$ and length ${\displaystyle L}$. To calculate the average linear mass density, ${\displaystyle {\overline{\lambda }}_m}$, of this one dimensional object, we can simply divide the total mass, ${\displaystyle M}$, by the total length, ${\displaystyle L}$: $${\displaystyle {\overline{\lambda }}_m=\frac{M}{L}}$$ If we describe the rod as having a varying mass (one that varies as a function of position along the length of the rod, ${\displaystyle l}$), we can write: $${\displaystyle m=m(l)}$$ Each infinitesimal unit of mass, ${\displaystyle dm}$, is equal to the product of its linear mass density, ${\displaystyle {\lambda }_m}$, and the infinitesimal unit of length, ${\displaystyle dl}$: $${\displaystyle dm={\lambda }_{m}dl}$$ 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, ${\displaystyle l}$) $${\displaystyle {\lambda }_m=\frac{dm}{dl}}$$

Units

Common units of linear mass density include:

• kilogram per meter (using SI base units)
• ounce (mass) per foot
• ounce (mass) per inch
• pound (mass) per yard: used in the North American railway industry for the linear density of rails
• pound (mass) per foot
• pound (mass) per inch

Textile applications

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 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 is measured.

Common units include:

• 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 (dtex), a unit for the linear density of fibers, defined as the mass in grams per 10,000 meters

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