Question
State and explain the laws of vibrating strings.
✓
Answer
The fundamental frequency of vibration of a stretched string or wire of uniform cross section is 1
$
n =\frac{1}{2 L} \sqrt{\frac{T}{m}}
$
where $L$ is the vibrating length, $m$ the mass per unit length (linear density) of the string and $T$ the tension in the string. From the above expression, we can state the following three laws of vibrating strings.
(1) Law of length : The fundamental frequency of vibrations of a stretched string is inversely proportional to its vibrating length, if the tension and mass per unit length are kept constant.
If $T$ and $m$ are constant,
$n \propto \frac{1}{L}$ or $nL =$ constant.
(2) Law of tension : The fundamental frequency of vibrations of a stretched string is directly proportional to the square root of the applied tension, if the length and mass per unit length are kept constant. If $L$ and $m$ are constant,
$n \propto$ or $\sqrt{T}$ or $n ^2 / T =$ constant.
(3) Law of mass (or law of linear density): The fundamental frequency of vibrations of a stretched string is inversely proportional to the square root of its mass per unit length, if the length and tension are kept constant. If $L$ and $T$ are constant,
$n \propto \frac{1}{\sqrt{m}}$ or $n ^2 m =$ constant.
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