# What is linear mass density?

Feb 13, 2014

Linear mass density is the amount of mass per unit length.

Just as ordinary density is mass per unit volume, linear density is mass per unit length. Linear densities are usually used for long thin objects such as strings for musical instruments.

EXAMPLE

Suppose we have a 0.80 mm diameter guitar string made of carbon steel (density = 7.860 g/cm³). What is the linear mass density of the string?

SOLUTION

Step 1: Find the volume of a length (say, 1 m = 100 cm) of the string.

$V = \text{πr"^2"l}$

The radius of the string is half the diameter.

$r = \text{0.80 mm"/2 = "0.40 mm" = "0.040 cm}$

$V = {\text{πr"^2"l" = π × ("0.040 cm")^2 × "100 cm" = "0.503 cm}}^{3}$

Step 2: Find the mass of the wire.

Mass =$\text{0.503 cm"^3 × "7.860 g"/("1 cm"^3) = "3.95 g}$

Step 3: Find the linear mass density.

Linear mass density = $\text{mass"/"length" = "3.95 g"/"1 m" = "4.0 g/m}$ (2 significant figures).