First slide

Stationary waves

Question

Two vibrating strings of the same material but length L and 2L have radii 2r and r respectively . They are stretched under the same tension. Both strings vibrates in their fundamental nodes. The one of length ‘L’ with frequency $n_{1}$ and other with frequency $n_{2}$ . The ratio of $n_{1} / n_{2}$ is given

Moderate

A

2
B

4
C

8
D

1

Solution

$F u n d a m e n t a l f r e q u e n c y f o r f i r s t s t r i n g : n_{1} = \frac{1}{2 l} \sqrt{\frac{T}{4 π r^{2} ρ}} \Rightarrow n_{1} = \frac{1}{4 l} \sqrt{\frac{T}{π r^{2} ρ}} \to (1) and F u n d a m e n t a l f r e q u e n c y f o r s e c o n d s t r i n g : n_{2} = \frac{1}{2 (2 l)} \sqrt{\frac{T}{π r^{2} ρ}} \to (2) D i v i d e i / i i \frac{n_{1}}{n_{2}} = 1$

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