Question

A wire with linear density \[3\text{ gm/mm}\]is used as a sonometer wire for producing a frequency of 50 Hz. This length of this wire is now halved, while the tension is reduced by $\dfrac{1}{4}\text{th}$ of its initial tension. What will be the frequency of vibrations produced:
A.) $10\text{ Hz}$
B.) $30\text{ Hz}$
C.) $50\text{ Hz}$
D.) $70\text{ Hz}$

Answer 1

Hint: The frequency produced in a wire is directly proportional to the square root of tension on the wire and inversely proportional to the square root of mass per unit length. So when the tension or the length of the wire is changed, the frequency changes.

Complete step by step answer:
Suppose we have a string of mass per unit length ‘m’ and tension applied to it is T. So, the frequency mode of the wave produced in this string for these parameters is,

$\text{Frequency (Hz)}=\dfrac{n}{2L}\sqrt{\dfrac{T}{m}}$

Where n is the nth harmonic of the frequency.

In the problem it is given that a particular frequency of 50 Hz is produced for an arbitrary mode for a tension of T and mass per unit length m. So, we can write

$50\text{Hz}=\dfrac{n}{2L}\sqrt{\dfrac{T}{m}}$ …. Equation (1)

So, in this problem we are reducing the tension to $\dfrac{1}{4}\text{th}$ of its initial tension and we are reducing the length to half. So, we can write that particular frequency ‘f’ as,

$f=\dfrac{n}{2\left[ \dfrac{L}{2} \right]}\sqrt{\dfrac{T/4}{m}}=\dfrac{n}{L}\sqrt{\dfrac{T}{4m}}\text{ }\left( \because m=\dfrac{M}{L} \right)$



Where M is the total mass of the string. We can rewrite the above equation as,

$f=\dfrac{n}{2L}\sqrt{\dfrac{T}{m}}$ ….. equation (2)
So, dividing equation (2) by equation (1), we get

$\dfrac{f}{50}=\dfrac{2}{2}$
$\Rightarrow f=\dfrac{50}{1}$
$\therefore f=50\text{ Hz}$

So, the frequency of the string does not change after we change the tension and length of the string.

So, the answer to the question is option (C)- 50 Hz.

Note: The frequency of the wave produced in a string of length L, tension T and linear mass density m will,

a.) Increase if the tension T is increased
b.) Decreases if the mass of the string (M) is increased
c.) Decreases if the length of the string is increased.

The frequency produced for different multiples of wavelength in the string is given by $\dfrac{v}{\left( \dfrac{2L}{n} \right)}$, where $v$ is the velocity of the wave in the string.
$\dfrac{2L}{n}$, where n is a positive number is the different modes of wavelength produced in the string.
The velocity of a wave in a string is given by $v=\sqrt{\dfrac{T}{m}}$, where T is the tension in the string and m is the mass per unit length.