Question

An all female guitar septet is getting ready to go on stage. The lead guitarist, Kira,who is always in tune, plucks her low E string and the other six members, sequentially, do the same. Each member records the initial beat frequency between her low E string and Kira's low E string.

To tune an instrument using beats, more information than just the beat frequency is needed. In addition to recording the initial beat frequency , each member, except Diane, also records the change in the frequency (increase or decrease) when they increase the tension in their low E string.

Rank each member on the basis of the initial frequency of their low E string.
Rank from largest to smallest. To rank items as equivalent, overlap them.

Aiko=3Hz, fbeat increases
Chandra=1Hz fbeat decreases
Evita=5Hz fbeat decreases.
Freja=3Hz fbeat decreases.
Buffy=4Hz fbeat increases.

Answer 1

Concepts and reason

The concept required to solve the given problem is beat frequency.

Initially, calculate the frequency of Aiko by using the expression of the beat frequency. Next, calculate the frequency of Chandra by using the expression of the beat frequency. After that, calculate the frequency of Evita by using the expression of the beat frequency. Later, calculate the frequency of Freja by using the expression of the beat frequency. Next, calculate the frequency of buffy by using the expression of the beat frequency. Finally, Rank the member based on initial frequency from largest to smallest by using the frequencies of the members.

Fundamentals

Beats occur when two waves of nearby frequencies overlap. The number of beats per second is equal to the difference in the frequency.

The expression for the beat frequency is,

${f_{{\rm{beat}}}} = \left| {{f_2} - {f_1}} \right|$

Here, ${f_1}$ is the frequency of the first wave and ${f_2}$ is the frequency of the second wave.

The expression for the fundamental frequency is,

$f = \frac{v}{{2L}}$

Here, v is the velocity of the wave and L is the length of the string.

The expression for the wave velocity is,

$v = \sqrt {\frac{T}{\mu }}$

Here, T is the tension and $\mu$ is the linear mass density.

Let f is the frequency of the Kira and ${f_1}$ is the frequency of the Aiko.

The beat frequency of Kira and Aiko is,

${f_{{\rm{beat}}}} = \left| {{f_2} - {f_1}} \right|$

Substitute f for ${f_2}$ and 3 Hz for ${f_{{\rm{beat}}}}$ .

$3\;{\rm{Hz}} = \left| {f - {f_1}} \right|$

From the above equation,

$\begin{array}{l}\\f - {f_1} = 3\;{\rm{Hz}}\\\\{f_1} - f = 3\;{\rm{Hz}}\\\end{array}$

From the above equations, in the first equation, beats decrease if ${f_1}$ increases and in the second equation, beats increase if ${f_1}$ increases.

From the given condition, beats increase. So, the beat frequency is,

$\begin{array}{c}\\{f_1} - f = 3\;{\rm{Hz}}\\\\{f_1} = f + 3\;{\rm{Hz}}\\\end{array}$

Let f is the frequency of the Kira and ${f_2}$ is the frequency of the Chandra.

The beat frequency of Kira and Chandra is,

$1\;{\rm{Hz}} = \left| {f - {f_2}} \right|$

From the given condition, beats decrease. So, the beat frequency is,

$\begin{array}{c}\\f - {f_2} = 1\;{\rm{Hz}}\\\\{f_2} = f - 1\;{\rm{Hz}}\\\end{array}$

Let f is the frequency of the Kira and ${f_3}$ is the frequency of the Evita.

The beat frequency of Kira and Evita is,

$5\;{\rm{Hz}} = \left| {f - {f_3}} \right|$

From the given condition, beats decrease. So, the beat frequency is,

$\begin{array}{c}\\f - {f_3} = 5\;{\rm{Hz}}\\\\{f_3} = f - 5\;{\rm{Hz}}\\\end{array}$

Let f is the frequency of the Kira and ${f_4}$ is the frequency of the Freja.

The beat frequency of Kira and Freja is,

$3\;{\rm{Hz}} = \left| {f - {f_4}} \right|$

From the given condition, beats decrease. So, the beat frequency is,

$\begin{array}{c}\\f - {f_4} = 3\;{\rm{Hz}}\\\\{f_4} = f - 3\;{\rm{Hz}}\\\end{array}$

Let f is the frequency of the Kira and ${f_5}$ is the frequency of the Buffy.

The beat frequency of Kira and Buffy is,

$4\;{\rm{Hz}} = \left| {f - {f_5}} \right|$

From the given condition, beats increase. So, the beat frequency is,

$\begin{array}{c}\\{f_5} - f = 4\;{\rm{Hz}}\\\\{f_5} = f + 4\;{\rm{Hz}}\\\end{array}$

The rank of the member based on initial frequency from largest to smallest is,

Buffy>Aiko>Chandra>Freja>Evita.

Ans:

The rank of the member based on initial frequency from largest to smallest is Buffy>Aiko>Chandra>Freja>Evita.