Question

A sonometer wire has a length l and tension T. If on reducing the tension to half of its original value and changing the length, the second harmonic becomes equal to the fundamental frequency of the first case, then the new length of the wire is
a) $l/\sqrt 2 $
b) $\sqrt 2 l$
c) $l/2$
d) $2l$

Answer 1

Hint: To solve this problem we will substitute the given values in the sonometer formula.
We need to use a sonometer working formula. i.e. \[{f_n} = \dfrac{n}{{2l}}\sqrt {\dfrac{T}{m}} \], where, \[{f_n}\]​ is the frequency of \[nth\]mode, \[n\] is the mode number, \[l\] is the length of the wire, \[T\] is the tension in the wire, \[m\] is the linear mass density or mass per unit length of the wire.

Complete step-by-step answer:
If the tension reduced to half, and length becomes \[{l_1}\]​ then from the above condition,
\[\dfrac{1}{{2l}}\sqrt {\dfrac{T}{m}} = \dfrac{2}{{2{l_1}}}\sqrt {\dfrac{T}{{2m}}} \]
As n is a constant value, we neglected n in the formula.
$ \Rightarrow {l_1} = 1\sqrt 2 $

Thus, the correct answer to this question is option (b).

Note: A sonometer is a device for demonstrating the relationship between the frequency of the sound produced by a plucked string, and the tension, length and mass per unit length of the string. These relationships are usually called Mersenne's laws after Marin Mersenne (1588-1648), who investigated and codified them. It is used to measure the tension, frequency or density of vibrations. In the field of medicine, it is used to test both hearing and bone density. The vibrations produced by the string works under the principle of resonance and is often represented as sinusoidal waves. This is very useful.