Linear density

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.

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

To calculate the average linear mass density,

, of this one dimensional object, we can simply divide the total 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,

, is equal to the product of its linear mass density,

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 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.

[1][2] Consider a long, thin wire of charge

To calculate the average linear charge density,

, of this one dimensional object, we can simply divide the total charge,

, is equal to the product of its linear charge density,

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).

In drawing or printing, 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 of a single random variable.