A sonometer wire supports a $4Kg$ load and vibrates in fundamental mode with a tuning fork of frequency $416Hz$ . The length of the wire between the bridges is now doubled. In order to maintain fundamental mode, the load should be changes to
A) $1Kg$
B) $2Kg$
C) $8Kg$
D) $16Kg$

Hint: Here in this question, we have to find the change of the load in fundamental mode for which the length of the wire is doubled between the bridges. This requires the usage of the fundamental frequency since it is at this frequency that the fundamental mode's load varies as the length is increased.

Formula Used:
Fundamental Frequency, ${f_ \circ } = \dfrac{V}{{4l}}$
Since, ${f_ \circ } = \dfrac{1}{{4l}}\sqrt {\dfrac{T}{\mu }} $
Formula of Sonometer, $T = mg$



Complete answer:

We know that, Fundamental Frequency,
${f_ \circ } = \dfrac{1}{{4l}}\sqrt {\dfrac{T}{\mu }} $
As according to the question, after increasing the length,
$l' = 2l$
As putting this value in Fundamental frequency equation, we get,
$f = \dfrac{1}{{4l'}}\sqrt {\dfrac{{T'}}{\mu }} $
After putting the value of $l' = 2l$ in the above equation, we get,
$f = \dfrac{1}{{4 \times 2l}}\sqrt {\dfrac{{T'}}{\mu }} $
In accordance with the query, it is implied that,
$f = {f_ \circ }$
By putting the value of both we get the following equation,
$\dfrac{1}{{4 \times 2l}}\sqrt {\dfrac{{T'}}{\mu }} = \dfrac{1}{{4l}}\sqrt {\dfrac{T}{\mu }} $
By further solution and using $T = mg$ in the above equation we get,
$\dfrac{1}{2}\sqrt {\dfrac{{m'g}}{\mu }} = \sqrt {\dfrac{{mg}}{\mu }} $
By doing solution by making them simple we get,
$\dfrac{1}{2}\sqrt {m'} = \sqrt m $
By taking square both side, we get,
$m' = 4 \times m$
By putting the value of m from the question we get,
$m' = 4 \times 4$
By completing the solution we get,
$m' = 16Kg$
Therefore, the correct answer for change of load in fundamental mode is $16Kg$ .


Hence, the correct option is (D).


Note: We all should know that, A stretched string's length, linear mass density, frequency, and tension are all essentially studied with a sonometer. Devices based on the resonance theory include sonometers. It is utilized to establish the tuning fork's frequency as well as the rules of vibration of stretched strings.