First slide

Stationary waves

Question

Two uniform strings A and B made of steel are made to vibrate under the same tension. if the first overtone of A is equal to the second overtone of B and if the radius of A is twice that of B, the ratio of the lengths of the strings is

Moderate

A

1 : 2
B

1 : 3
C

1 : 4
D

1 : 6

Solution

First overtone of string A = Second overtone of string B.
$\Rightarrow$ Second harmonic of A = Third harmonic of B
$\Rightarrow n_{2} = n_{3} \Rightarrow {[2 (n_{1})]}_{A} = {[3 (n_{1})]}_{B}$ ( $∵ n_{1} = \frac{1}{2 l} \sqrt{\frac{T}{{πr}^{2} ρ}}$ )
$\Rightarrow 2 [\frac{1}{2 l_{A} r_{A}} \sqrt{\frac{T}{πρ}}] = 3 [\frac{1}{2 l_{B} r_{B}} \sqrt{\frac{T}{πρ}}]$
$\frac{l_{A}}{l_{B}} = \frac{2}{3} \frac{r_{B}}{r_{A}} \Rightarrow \frac{l_{A}}{l_{B}} = \frac{2}{3} \times \frac{r_{B}}{(2 r_{B})} = \frac{1}{3}$

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