In order to double the frequency of the fundamental note emitted by a stretched string, the length is…

In order to double the frequency of the fundamental note emitted by a stretched string, the length is reduced to 3/4th of the original length and the tension is changed. The factor by which the tension is to be changed, is
  1. $\frac{3}{8}$
  2. $\frac{2}{3}$
  3. $\frac{8}{9}$
  4. $\frac{9}{4}$

Solution

Frequency, $n = \frac{1}{2l}\sqrt{\frac{T}{\mu}} \propto \frac{\sqrt{T}}{l}$ $\Rightarrow \frac{T_2}{T_1} = \left(\frac{n_2}{n_1}\right)^2 \left(\frac{l_2}{l_1}\right)^2 = (2)^2 \left(\frac{3}{4}\right)^2 = \frac{9}{4}$

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