Problem 43 - College Physics

Question

A steel wire with mass \(25.0 \mathrm{~g}\) and length \(1.35 \mathrm{~m}\) is strung on a bass so that the distance from the nut to the bridge is \(1.10 \mathrm{~m}\). (a) Compute the linear density of the string. (b) What velocity wave on the string will produce the desired fundamental frequency of the \(\mathrm{E}_{1}\) string, \(41.2 \mathrm{~Hz}\) ? (c) Calculate the tension required to obtain the proper frequency. (d) Calculate the wavelength of the string's vibration. (e) What is the wavelength of the sound produced in air? (Assume the speed of sound in air is \(343 \mathrm{~m} / \mathrm{s}\).)

Step-by-Step Solution

Answer

The linear density of the string is \(0.0185 kg/m\). The velocity of the wave on the string should be \(91.52 m/s\). The tension required to obtain the proper frequency is \(154 N\). The wavelength of vibration on the string is \(2.2 m\) and the wavelength of the sound produced in air is \(8.33 m\).

Step 1: Compute the Linear Density

The linear density \(\mu\) of the string is defined as the mass divided by the length. This can be calculated using the formula: \(\mu = \frac{m}{L}\), where \(m\) is the mass of the string (25.0g = 0.025kg) and \(L\) is the full length of the string (1.35m).

Step 2: Calculate the Wave Velocity

The wave velocity \(v\) on the string can be computed using the equation for the fundamental frequency of a standing wave on a string: \(f = \frac{v}{2L}\) where \(L\) is the distance from the nut to the bridge (1.10m), \(f\) is the desired fundamental frequency (41.2Hz). We rearrange the equation to solve for \(v\) as \(v = 2fL\).

Step 3: Determine the Tension Required

The tension \(T\) required to obtain the proper frequency can be computed using the formula for wave speed on a string under tension: \(v = \sqrt{\frac{T}{\mu}}\), where \(\mu\) is the linear density we determined in step 1 and \(v\) is the wave velocity we found in Step 2. We rearrange the equation to solve for \(T\) as \(T = v^2*\mu\).

Step 4: Calculate the Wavelength of the String's Vibration

Since this is a standing wave, the wavelength \(\lambda\) of the string's vibration is twice the length of the string \(L\), thus \(\lambda = 2L\). Note that \(L\) here is the distance between the nut and the bridge (1.1m).

Step 5: Determine the Wavelength of the Sound Produced in Air

The wavelength of the sound produced in air can be calculated using the wave speed equation \(v = f\lambda\), with \(v\) being the speed of sound in air (343 m/s), \(f\) being the frequency (41.2 Hz). We rearrange the equation to solve for \(\lambda\) as \(\lambda = \frac{v}{f}\).

Key Concepts

Linear Density

Linear density is a crucial concept in understanding how waves travel through a string. It describes the mass of the string per unit length. This is important because it affects how fast a wave can move along the string. To calculate the linear density (\(\mu\)), you can use the formula:

• \(\mu = \frac{m}{L}\)

where \(m\) is the mass of the string and \(L\) is the total length of the string. In this case, with a steel wire mass of \(25.0\) grams, equivalent to \(0.025\) kg, and a length of \(1.35\) meters, the linear density is \(\mu = \frac{0.025\, \text{kg}}{1.35\, \text{m}}\). This gives us the linear density of the string in units of kg/m.Understanding linear density helps predict how different strings might affect sound production in musical instruments. Strings with higher linear density typically produce a deeper sound, given the same tension and wave velocity.

Wave Velocity

Wave velocity on a string is a measure of how quickly waves travel along it. This is paramount to musical instruments like guitars since it correlates directly to pitch. The formula that links wave velocity \(v\) to frequency \(f\) and length \(L\) of the string for the fundamental frequency is:

• \(f = \frac{v}{2L}\)

By rearranging it, we can solve for wave velocity:

• \(v = 2fL\)

Here, the fundamental frequency of the note is given as \(41.2\) Hz, and the length \(L\) (distance from nut to bridge) is \(1.10\) meters. Plug these values into the equation to find the wave velocity. This relationship demonstrates the dynamic nature of sound, and how changing one factor can affect the others. A change in string length or frequency can alter the wave velocity, influencing the sound produced.

String Tension

String tension is crucial for musical instruments, as it affects the pitch of the notes they produce. Greater tension generally raises the frequency, producing higher-pitched sound. The formula used to calculate the required tension \(T\) involves the wave velocity \(v\) and the linear density \(\mu\):

• \(v = \sqrt{\frac{T}{\mu}}\)

Rearranging this equation provides a formula for tension:

• \(T = v^2 \cdot \mu\)

By using the calculated wave velocity and known linear density, you can find the tension needed to produce the desired frequency. For musicians, adjusting the tension by tuning a string ensures that their instrument is pitch-perfect. Tension is usually adjusted by turning tuning pegs, changing the note produced to match the desired pitch or frequency.

Standing Waves

Standing waves are a central concept in understanding how musical instruments generate sound. In the scenario of a string fixed at both ends, like a guitar string, a standing wave forms when waves reflect between these points, leading to a pattern of nodes and antinodes.The wavelength \(\lambda\) in standing waves is related to the length of the string, where the fundamental frequency has a wavelength:

• \(\lambda = 2L\)

For this string with length \(L = 1.10\) meters, the wavelength is \(\lambda = 2 \times 1.10\) m, leading to \(2.20\) meters.Understanding standing waves allows musicians and engineers to design and use instruments effectively, ensuring consistent and predictable sound output. When the wavelength matches the length of the string, it results in a clear and harmonious tone, enriching the musical experience.