Question

A particular steel guitar string has mass per unit length of \(1.93 \mathrm{~g} / \mathrm{m}\). a) If the tension on this string is \(62.2 \mathrm{~N},\) what is the wave speed on the string? b) For the wave speed to be increased by \(1.0 \%\), how much should the tension be changed?

Short answer

Answer: To increase the wave speed of the steel guitar string by 1%, the tension should be changed by approximately 2.07 N.

Step-by-step solution

(Step 1: Find the wave speed under the given tension)

First, we need to convert the mass per unit length value to base SI units (kilograms per meter) so that our calculations are consistent. \(1.93 \mathrm{~g/m} = \frac{1.93}{1000} \mathrm{~kg/m} = 0.00193 \mathrm{~kg/m}\). The wave speed formula is \(v = \sqrt{\frac{T}{\mu}}\). Plug in the given tension \(T = 62.2\mathrm{~N}\) and mass per unit length \(\mu = 0.00193 \mathrm{~kg/m}\), then calculate the wave speed \(v\): \(v = \sqrt{\frac{62.2}{0.00193}} \approx 180.87 \mathrm{~m/s}\) The wave speed on the string under the given tension is approximately \(180.87 \mathrm{~m/s}\).

(Step 2: Find the tension change for a 1% increase in wave speed)

Now we need to calculate the tension change required to increase the wave speed by 1%. First, determine the new wave speed: New wave speed \(v' = v + 0.01v = 1.01v \approx 1.01 \cdot 180.87 \approx 182.68 \mathrm{~m/s}\) Now, use the wave speed formula again, but this time with the new wave speed \(v'\). We will solve for the new tension \(T'\): \(182.68 = \sqrt{\frac{T'}{0.00193}} \Rightarrow (182.68)^2 = \frac{T'}{0.00193} \Rightarrow T' \approx 64.27 \mathrm{~N}\) Finally, we will find the tension change: \(\Delta T = T' - T \approx 64.27 - 62.2 = 2.07 \mathrm{~N}\) To increase the wave speed by 1%, the tension should be changed by approximately \(2.07 \mathrm{~N}\).

Key concepts

Tension in Strings

Understanding tension in strings is crucial for explaining wave motion. Tension can be thought of as the 'stretching force' exerted by a string or cable when it is pulled tight. This force is always directed along the length of the string and can significantly influence the speed of a wave traveling through it.

In the context of a musical instrument like a guitar, the tension in the strings can be adjusted by tuning pegs, which either tighten or loosen the string. This adjustment changes the frequency of the sound produced when the string is plucked. A more tightened string, holding greater tension, vibrates at a higher frequency and thus produces a higher note.

The formula for wave speed (\(v\text{, measured in meters per second}\)) in a string under tension is given by \( v = \sqrt{\frac{T}{\mu}} \) where:\

\
• \(T\text{, measured in newtons (N)}\) is the tension in the string.\
\
• \(\mu\text{, measured in kilograms per meter (kg/m)}\) is the mass per unit length.\
\

So, altering the tension affects the wave speed; increasing tension typically results in a faster wave speed and hence a higher pitch.

Mass Per Unit Length

The mass per unit length, often represented by the Greek letter mu (\(\mu\)), is a property that helps define how waves travel through a medium. Specifically, it is the mass measured in kilograms of a uniform object, like a string, divided by its length in meters.

This property affects the wave dynamics in that a greater mass per unit length means that there is more mass that needs to be moved when a wave passes through. As a result, this tends to slow down the wave speed. It is important because it represents the inertia of the string section that must be overcome by the tension force during wave propagation.

In musical terms, thinner strings have a smaller mass per unit length and therefore can vibrate more quickly, producing higher frequencies, whereas thicker strings vibrate more slowly due to their greater mass per unit length, resulting in lower frequencies.

Velocity of Wave

The velocity or speed of a wave on a string is the rate at which the wave travels through the medium. The wave speed formula given by \( v = \sqrt{\frac{T}{\mu}} \) succinctly ties together two major properties: the tension in the string and the mass per unit length of the string.

To see the dynamics of wave velocity at play, consider the exercise provided. With an increase in tension, holding mass per unit length constant, the square root relationship implies that wave speed increases too. This is evident when we compare the original wave speed of approximately \(180.87 \mathrm{~m/s}\) under a tension of \(62.2 \mathrm{~N}\) to the new wave speed necessary for a 1% increase.

The tweak in tension yields a noticeable difference in the wave's behavior—a higher wave speed. This principle is practically applied in musical instruments to change pitch and in engineering to control vibrations in structures like bridges and aerial cables.