The frequency f of vibration of a violin string is inversely proportional to its length L. The constant of proportionality k is positive and depends on the tension and density of the string.

(A) write an equation that represents this variation.
(B) what effect does doubling the length on the string have on the frequency of its vibration?

We have been given that the frequency f of vibration of a violin string is inversely proportional to its length L. The constant of proportionality k is positive and depends on the tension and density of the string.

(A) We know that two inversely proportional quantities are in form [tex]y=\frac{k}{x}[/tex], where y is inversely proportional to x and k is constant of proportionality.

Upon substituting our given values, we will get:

[tex]f=\frac{k}{L}[/tex]

Therefore, our required equation would be [tex]f=\frac{k}{L}[/tex].

(B) For part, we have been given that length is twice, so our new frequency will be [tex]f_n[/tex] and new length [tex]L_n[/tex] is [tex]2L[/tex].

Upon substituting [tex]2L[/tex] in our equation as:

[tex]f_n=\frac{k}{L_n}[/tex]

[tex]f_n=\frac{k}{2L}[/tex]

[tex]f_n=\frac{1}{2}\cdot \frac{k}{L}[/tex]

[tex]f_n=\frac{1}{2}\cdot f[/tex]

Upon comparing [tex]f_n[/tex] with [tex]f[/tex], we can see that [tex]f_n[/tex] is half the value of [tex]f[/tex].

Therefore, the frequency of vibration of violin gets half, when we double the length of the string.