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**[SOLVED] Change in Tension & Fundamental Frequency of a String**

**Problem.**Show that if the tension in a streched string is change by a small amount [itex]\Delta F_T[/itex], the frequency of the fundamental is changed by a small amount [itex]\Delta f = 1/2 (\Delta F_T / F_T) f[/itex].

Let T be the intial tension and h the change in tension. The velocity of a transverse wave on the string is [itex]v = \sqrt{T/\mu}[/itex]. The initial frequency is

[tex]f = \frac{v}{\lambda} = \frac{\sqrt{T}}{\lambda \sqrt{\mu}}[/tex]

The new frequency f' is

[tex]f' = \frac{v'}{\lambda} = \frac{\sqrt{T + h}}{\lambda \sqrt{\mu}}[/tex]

The difference is:

[tex]f' - f = \frac{1}{\lambda \sqrt{\mu}} \, (\sqrt{T + h} - \sqrt{T})[/tex]

That looks nothing like what I'm trying to show. Now, if I multiply the RHS by [itex]\sqrt{T} / \sqrt{T}[/itex], I get

[tex]f' - f = \frac{\sqrt{T + h} - \sqrt{T}}{\sqrt{T}} \, f[/tex]

and if I do it again, I get

[tex]f' - f = \frac{\sqrt{T(T + h)} - T}{T} \, f[/tex]

which is as close as I could get to what needs to be shown.

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