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Superposition of Waves — Physics STD 12 Science — Question

Maharashtra BoardEnglish MediumSTD 12 SciencePhysicsSuperposition of Waves4 Marks
Question
How does the fundamental frequency of a vibrating string depend on the radius of cross section of the string and the mass density of the material of the string?
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Answer

Consider a string stretched between two rigid supports a distance $L$ apart. Let $T$ be the tension in the string, $r$ be its radius of cross section and $p$ be the mass density of its material. Then, the mass of the string $M=\left(\pi r^2 L\right) p$, so that its linear density, i.e., mass per unit length, $m=M / L=\pi r^2 p$.According to the law of mass of a vibrating string, the fundamental frequency $(n)$ is inversely proportional to the square root of its linear density, when $T$ and $L$ are constant.
$ n \propto \frac{1}{\sqrt{m}}$
$\therefore n \propto \frac{1}{\sqrt{\pi r^2 \rho}}$
$\therefore n \propto \frac{1}{r} \text { when } L, T \text { and } \rho \text { are constant, and }$
$n \propto \frac{1}{\sqrt{\rho}} \text { when } L, T \text { and } r \text { are constant. }$
Question $63.$
A string/wire is stretched between two rigid supports. State any two factors on which the fundamental frequency of the string/wire depends.
Answer:
1. Tension in the string/wire
2. length of the string/wire (or radius or mass per unit length or mass density of the material of the string/wire)

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