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------------------------------------------------ A violin with 5 strings (all of the same material) is tuned to a perfect fifth (3:2). Each string is under the same tension. Given this information, what is the difference in width of each string?

The difference in width of each string on the violin can be calculated using the formula for the fundamental frequency of a vibrating string, which is proportional to the square root of the tension divided by the linear mass density. Let's assume the linear mass density (mass per unit length) of the strings is the same for all strings.

The formula for the fundamental frequency of a vibrating string is:

f = (1/2L)*√(T/m)

Where:

f is the frequency (related to pitch),
L is the length of the string,
T is the tension in the string, and
m is the linear mass density of the string.

In this case, all strings are under the same tension, and we want to find the difference in width (linear mass density) of each string. Let's denote the width of the thinnest string as W

For strings with a perfect fifth interval (3:2 frequency ratio), the frequencies are in a 2:3 ratio. Therefore, we can set up the following relationship between two adjacent strings:

f₁/f₂ = 3/2

We can write then:

2(1/2L₁)*√(T/m₁) = 3*(1/2L₂)*√(T/m₂)

We assueme the same tension and the same length for both strings, then:

√(m₂/m₁) = 3/2

m₂ = (9/4)m₁

So the linear mass density is 9/4 times smaller than that of the thicker string. This represents the difference in width of each string.

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