Linear Density Calculator

Calculate linear density, wave speed, tension, and wire mass from mass, length, diameter, material density, or AWG gauge for strings and cables.

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Linear Density Formula

μ = M / L

• μ = linear density (kg/m)
• M = mass (kg)
• L = length (m)

For a cylindrical cross-section, linear density follows directly from volumetric density and diameter:

μ = ρ · π · d² / 4

In wave mechanics, linear density governs wave propagation speed on a stretched string under tension T:

v = √(T / μ)

Linear Density Definition

Linear density is mass per unit length, symbolized μ (mu) in physics. The SI unit is kg/m. It is the standard measure for objects where cross-sectional area is negligible compared to length: wires, strings, ropes, textile fibers, and structural rods. It should be distinguished from linear charge density (charge per unit length, C/m) used in electrostatics, and from crystallographic linear density (atoms per unit length along a direction vector) used in materials science.

Reference: Linear Density of Common Objects

Wire values calculated from μ = ρπd²/4 using material density. Guitar string values from standard published sets.

Textile Fiber Units

Fiber diameters (10 to 30 μm) are too small and too irregular to measure directly, so the textile industry uses linear density as the primary size specification. Three unit systems are in use globally:

Conversion: 1 tex = 9 denier = 10 dtex. Typical benchmarks: fine merino wool ~18 D, standard polyester apparel ~75 D, denim warp yarn ~300 D, nylon carpet fiber ~1500 D.

Crystallographic Linear Density

In materials science, linear density has a distinct meaning: the number of atom centers lying on a direction vector divided by the length of that vector. This measures atomic packing efficiency along specific crystallographic directions. Close-packed directions have the highest linear density and are the preferred slip directions during plastic deformation, directly governing ductility and yield strength. In BCC metals (iron, tungsten, chromium) the close-packed direction is the body diagonal <111>; in FCC metals (copper, aluminum, gold) it is the face diagonal <110>. This calculator computes mass-based linear density only.

Wave Speed and Guitar Strings

The roughly 18:1 ratio in linear density between a standard low E string (5.8 x 10^-3 kg/m) and a high E string (3.1 x 10^-4 kg/m) is what allows both strings to operate at playable tension while spanning a two-octave frequency range. Since f = v/(2L) and v = sqrt(T/mu), a lower mu produces a higher frequency at the same tension and scale length. Wound strings add mass via a wrap wire without adding significant stiffness, keeping the string flexible while achieving the high linear density required for bass notes.

Is linear density constant?

Linear density is constant only when mass is uniformly distributed along the length. A uniform wire or rod has constant μ at every cross-section. A tapered shaft, a wound string, or a composite beam has a position-dependent μ(x); the value μ = M/L gives only the average over the whole length.

Does tension affect linear density?

Yes. Tension stretches an object, increasing length while mass stays fixed, so μ decreases slightly. For steel at working loads this is negligible (a 0.1% elongation reduces μ by 0.1%). For highly elastic materials such as rubber bands, the effect is significant and must be accounted for.

How to Measure Linear Density

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