Question

A metal guitar string has a linear mass density of $\mu=3.20 \mathrm{g} / \mathrm{m} .$ What is the speed of transverse waves on this string when its tension is \(90.0 \mathrm{N} ?\)

Short answer

Answer: The speed of transverse waves on the guitar string is approximately 168 m/s.

Step-by-step solution

Convert linear mass density to kg/m

First, we need to convert the given linear mass density from grams per meter (g/m) to kilograms per meter (kg/m). To do this, divide the given value by 1000: $$ \mu = \frac{3.20\,\mathrm{g/m}}{1000} = 0.00320\,\mathrm{kg/m} $$

Calculate the speed of transverse waves

Now that we have the linear mass density in kg/m, we can use the formula for the speed of transverse waves on a string: $$ v = \sqrt{\frac{T}{\mu}} $$ Plug in the values for tension (\(T=90.0\,\mathrm{N}\)) and linear mass density (\(\mu=0.00320\,\mathrm{kg/m}\)): $$ v = \sqrt{\frac{90.0\,\mathrm{N}}{0.00320\,\mathrm{kg/m}}} $$ Calculate the result: $$ v \approx 168\,\mathrm{m/s} $$

State the result

The speed of transverse waves on the guitar string when its tension is \(90.0\,\mathrm{N}\) is approximately \(168\,\mathrm{m/s}\).