Question

Fundamental frequency of a sonometer wire is $n$. If the length and diameter of the wire are doubled keeping the tension same, then the new fundamental frequency is ?

Answer 1

First calculate the frequency of sonometer wire at its original length, diameter and tension. After that calculate the frequency of sonometer wire when length, diameter is doubled tension is same. Compare both the frequency to get a relation between them.

Complete step by step answer:
Let the initial length of the wire be l, diameter be d and the tension on the wire be T, then the frequency is given by
$n = \dfrac{1}{{2l}}\sqrt {\dfrac{T}{{\dfrac{m}{l}}}}$
We are having linear mass density in the denominator as wire is a one-dimension object therefore, we can consider it as a cylinder. When wire of length l and diameter is doubled
$n' = \dfrac{1}{{2(2l)}}\sqrt {\dfrac{{T(2l)}}{{\pi {4^2}(2l)}}}$
$\Rightarrow n' = \dfrac{1}{{2(2l)}} \times \dfrac{1}{2}\sqrt {\dfrac{T}{{\dfrac{m}{l}}}}$
$\Rightarrow n' = \dfrac{1}{4} \times \dfrac{1}{{2l}}\sqrt {\dfrac{T}{{\dfrac{m}{l}}}}$
$\therefore n' = \dfrac{n}{4}$

Hence, the new fundamental frequency is $\dfrac{n}{4}$.

Additional Information:
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. Frequency is defined as the number of oscillations or occurrences per unit time. The unit of frequency is Hertz. Frequency is also defined as the reciprocal of time period. From the above calculation you can see that by increasing the length and diameter of the cross-section of wire the frequency decreases.

Note: For problems like this, first calculate frequency in the original case, and then calculate the frequency for the changed case. Compare both of them to get a relationship between them. You can also calculate the time period because the time period is reciprocal of frequency.
${\text{Time period = }}\dfrac{1}{{frequency}}$
When the tension on the wire is large the wire will break. This breaking point is called breaking stress. So breaking stress is defined as the force on the wire per unit cross-sectional area.
When the tension on the wire is large the wire will break. This breaking point is called breaking stress. So breaking stress is defined as the force on the wire per unit cross-sectional area.