Question
State the laws of vibrating strings
✓
Answer
The fundamental frequency of a vibrating string under tension is given as, $n =\frac{1}{2 l} \sqrt{\frac{ T }{ m }}$
From this formula, three laws of vibrating string can be given as follows:
i. Law of length: The fundamental frequency of vibrations of a string is inversely proportional to the length of the vibrating string if tension and mass per unit length are constant.
$\therefore n \alpha \frac{1}{l}$ .(if $T$ and $m$ are constant.)
ii. Law of tension: The fundamental frequency of vibrations of a string is directly proportional to the square root of tension if the vibrating length and mass per unit length are constant.
$\therefore n \alpha \sqrt{ T }$
(if I and $m$ are constant.)
iii. Law of linear density: The fundamental frequency of vibrations of a string is inversely proportional to the square root of mass per unit length (linear density), if the tension and vibrating length of the string are constant.
$\therefore n \alpha \frac{1}{\sqrt{ m }}$ (if $T$ and I are constant.)
If $r$ is the radius and $r$ is the density of the material of string, linear density is given as
Linear density $=$ mass per unit length
$=$ volume per unit length $\times$ density
$ =\frac{\pi r^2 l}{l} \times \rho$
$=\pi r^2 \rho$
As $n \propto \frac{1}{\sqrt{ m }}$, if $T$ and I are constant, we have,
$n \propto \frac{1}{\sqrt{\pi r^2 \rho}}$
i.e., $n \propto \frac{1}{\sqrt{\rho}}$ and $n \propto \frac{1}{ r }$
Thus the fundamental frequency of vibrations of a stretched string is inversely proportional to the radius of string and the square root of the density of the material of vibrating string.
From this formula, three laws of vibrating string can be given as follows:
i. Law of length: The fundamental frequency of vibrations of a string is inversely proportional to the length of the vibrating string if tension and mass per unit length are constant.
$\therefore n \alpha \frac{1}{l}$ .(if $T$ and $m$ are constant.)
ii. Law of tension: The fundamental frequency of vibrations of a string is directly proportional to the square root of tension if the vibrating length and mass per unit length are constant.
$\therefore n \alpha \sqrt{ T }$
(if I and $m$ are constant.)
iii. Law of linear density: The fundamental frequency of vibrations of a string is inversely proportional to the square root of mass per unit length (linear density), if the tension and vibrating length of the string are constant.
$\therefore n \alpha \frac{1}{\sqrt{ m }}$ (if $T$ and I are constant.)
If $r$ is the radius and $r$ is the density of the material of string, linear density is given as
Linear density $=$ mass per unit length
$=$ volume per unit length $\times$ density
$ =\frac{\pi r^2 l}{l} \times \rho$
$=\pi r^2 \rho$
As $n \propto \frac{1}{\sqrt{ m }}$, if $T$ and I are constant, we have,
$n \propto \frac{1}{\sqrt{\pi r^2 \rho}}$
i.e., $n \propto \frac{1}{\sqrt{\rho}}$ and $n \propto \frac{1}{ r }$
Thus the fundamental frequency of vibrations of a stretched string is inversely proportional to the radius of string and the square root of the density of the material of vibrating string.
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