Question

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

Answer 1

We know that when the string is stretched and plucked, it will start vibrating. The vibration in the string will provide a fundamental frequency in the string. The fundamental frequency is defined as the lowest frequency of the periodic waveform. The formula of the fundamental frequency is shown below.

Formula used:
The formula of the fundamental frequency of the string is given by
$f = \dfrac{1}{{2l}}\sqrt {\dfrac{T}{\mu }}$
Here, $f$ is the frequency, $l$ is the length of the string, $T$ is the tension produced in the string and $\mu$ is the linear density of the string.

Complete step by step answer:
When a string is stretched between two points and is plucked, it will start vibrating. The vibration of the string will produce a fundamental frequency that has nodes at the end points. There is a general that will be used to calculate the fundamental frequency of the string, according to the tension, length and mass of the string and is given by
$f = \dfrac{1}{{2l}}\sqrt {\dfrac{T}{\mu }}$

Now, when the length of the string is reduced to $\dfrac{3}{4}$ , the tension in the string will become $T'$ and the frequency of the string will be $f'$ and is given by
$f' = \dfrac{1}{{2\left( {\dfrac{3}{4}l} \right)}}\sqrt {\dfrac{{T'}}{\mu }}$
$\Rightarrow \,f' = \dfrac{2}{{3l}}\sqrt {\dfrac{{T'}}{\mu }}$
Now, as given in the question, we have to double the fundamental frequency and is shown below
$f' = 2f$
$\Rightarrow \,\dfrac{2}{{3l}}\sqrt {\dfrac{{T'}}{\mu }} = 2\left( {\dfrac{1}{{2l}}\sqrt {\dfrac{T}{\mu }} } \right)$
$\Rightarrow \,\sqrt {T'} = \dfrac{3}{2}\sqrt T$
Now, squaring the both sides, we get
$\therefore T' = \dfrac{9}{4}T$
Therefore, the factor by which the tension is to be changed is $\dfrac{9}{4}$ .

Hence, option D is the correct option.

Note: As we know that when the string is stretched and vibrated, a tension will be produced in it. Here, $T$ is the tension of the stretched string and $T'$ is the tension in the string when the length will be reduced to $\dfrac{3}{4}$ . When the parameters will change, there will be a change in the frequency of the string.