Linear Density Formula

Linear mass density formula, piece by piece

For a uniform wire or rod, μ is the same along its whole length. If you know the ordinary density ρ and the cross section area A, the mass in one metre is $\rho A \times 1$, so

$$\mu = \rho A = \rho \pi r^2$$

for a round wire of radius r. A thicker string of the same material therefore has a larger μ. A non uniform rod has a μ that changes from point to point, written μ(x), and its total mass is found by adding up μ times each small length.

Linear density formula in worked problems

Why μ matters for waves and music

The speed of a transverse wave on a string is $v = \sqrt{T/\mu}$. A heavy string carries waves more slowly, so with the same length and tension it vibrates at a lower frequency. That is why the bass strings of a guitar are thick and wound with extra wire: the extra mass per metre lowers their pitch without making them floppy. The fundamental frequency of a string fixed at both ends is

$$f_1 = \frac{1}{2L}\sqrt{\frac{T}{\mu}}$$

The wave speed itself is covered on the wave velocity formula page and the pitch link on the frequency formula page.

Units, dimensions and other uses

Mass over length gives [M¹ L⁻¹ T⁰] and the unit kg/m. In textiles, yarn thickness is quoted as linear density too, in units such as grams per kilometre. Engineers use mass per metre when choosing cables and rails. With charge in place of mass, the same idea gives the linear charge density formula, $\lambda = \frac{q}{L}$, in C/m. For a solid block, mass over volume gives ordinary density instead.

Mistakes to avoid

• Using volume density in a string wave problem. v = √(T/μ) needs kg/m, not kg/m³.
• Using the full spool mass for a cut length. μ stays the same, but m must match the L you use.
• Mixing up μ and λ. In wave questions λ is usually wavelength, so write μ for mass per length.
• Forgetting to square the radius in μ = ρπr².