On a guitar, the lowest toned string is usually strung to the E note, which prod

Question

On a guitar, the lowest toned string is usually strung to the E note, which produces sound at 82.4 Hz. The diameter of E guitar strings is typically 0.0500 inches and the scale length between the bridge and nut (the effective length of the string) is 25.5 inches Various musical acts tune their E strings down to produce a "heavier" sound or to better fit the vocal range of the singer. As a guitarist you want to detune the E on your guitar to B (62.8 Hz). If you were to maintain the same tension in the string as with the E string, what diameter of string would you need to purchase to produce the desired note? Assume all strings available to you are made of the same material. Number inches Unfortunately, none of the strings in your collection have such a large diameter. In fact, the largest diameter you possess is 0.05780 inches. If the tension on your existing string is denoted Tbefore, by what fraction will you need to detune (that is, lower the tension) of this string to achieve the desired B note? Number T. after before

Explanation / Answer

f=(1/2L)(T/)frequency is proportional to (1/)

is proportional to the square of the radius

82.4Hz --> 62.8Hz ,( 0.05inch -->??)

62.8/82.4=(1/)

1/= 0.580851

= 1/0.580851= r^2/0.025^2

r^2= 0.025^2/0.580851

r= 0.0328

diameter= 2r= 0.0656 inches

2nd part

L = length of string

T = tension

= linear density (mass per unit length)

The frequency is given by f = (1/(2L))(T/)

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For 2 different strings we can write:

f = (1/(2L))(T/)

f = (1/(2L))(T/)

Squaring gives:

f² = (1/(4L²))(T/)

f² = (1/(4L²))(T/)

Since L = L, taking the ratios of the equations gives:

(f/f)² = (T/T)(/) (equation 1)

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f/f = 82.4/62.8 (work out the value)

is proportional to cross-sectional area squared, hence is proportional to diameter squared:

/ = (0.05780/0.0500)² (work out the value)

Plug in these values into equation 1. Then you can find T/T.

T/T = 1.288

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If the initial tension is T and it is lowered by a fraction k (which is what the question asks for) then the final tension is T - kT = T(1 - k). So we have

T = T(1 - k)

k = 1 - (T/T)

T/T is simply the inverse of T/T. So you can work out k.

k = 1 – (1/1.288)

= 0.2238

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