Wave Motion 2 Question 20
20. Two vibrating strings of the same material but of lengths $L$ and $2 L$ have radii $2 r$ and $r$ respectively. They are stretched under the same tension. Both the strings vibrate in their fundamental modes. The one of length $L$ with frequency $v _1$ and the other with frequency $v _2$. The ratio $v _1 / v _2$ is given by
(2000, 2M)
(a) 2
(b) 4
(c) 8
(d) 1
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Answer:
Correct Answer: 20. (d)
Solution:
- Fundamental frequency is given by
$$ v=\frac{1}{2 l} \sqrt{\frac{T}{\mu}} $$
(with both the ends fixed)
$\therefore$ Fundamental frequency
$$ v \propto \frac{1}{l \sqrt{\mu}} \quad \text { (for same tension in both strings) } $$
where, $\mu=$ mass per unit length of wire
$$ \begin{array}{ll} & =\rho \cdot A \quad(\rho=\text { density }) \\ & =\rho\left(\pi r^{2}\right) \quad \text { or } \quad \sqrt{\mu} \propto r \\ \therefore \quad & \quad v \propto \frac{1}{r l} \\ \therefore \quad & \quad \frac{v _1}{v _2}=\left(\frac{r _2}{r _1}\right)\left(\frac{l _2}{l _1}\right)=\left(\frac{r}{2 r}\right)\left(\frac{2 L}{L}\right)=1 \end{array} $$
