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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 :
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Verified Answer
The correct answer is:
$1: 3$
Fundamental frequency of string,
$n=\frac{1}{2 l} \sqrt{\frac{T}{m}}$
where, $m=$ mass per unit length of the string
$=\pi r^2 \rho$
$\therefore \quad n=\frac{1}{2 l} \sqrt{\frac{T}{\pi r^2 \rho}}$
For string $A$ first overtone,
$n_A=\frac{1}{l_1} \sqrt{\frac{T_1}{m_1}}=\frac{1}{l_1} \sqrt{\frac{T_1}{\pi r_1^2 \rho}}$
For string $B$ second overtone,
$n_B=\frac{3}{2 l_2} \sqrt{\frac{T_2}{\pi r_2^2 \rho_2}}$
According to question,
$n_A=n_B$
$\frac{1}{l_1} \sqrt{\frac{T_1}{\pi r_1^2 l_1}}=\frac{3}{2 l_2} \sqrt{\frac{T_2}{\pi r_2^2 l_2}}$
but $\quad T_1=T_2$
$\therefore \quad \frac{2 l_2}{3 l_1}=\sqrt{\frac{\pi r_1^2 \rho}{\pi r_2^2 \rho}}$
$\frac{4 l_2^2}{9 l_1^2}=\frac{r_1^2}{r_2^2}$
$\left(\frac{l_1}{l_2}\right)^2=\frac{4 r_2^2}{9 r_1^2}$
$\frac{2 l_2}{3 l_1}=\frac{r_1}{r_2}$
$\frac{l_2}{l_1}=\frac{3 r_1}{2 r_2}$
$\frac{l_1}{l_2}=\frac{2 r_2}{3 r_1}$
but $\quad r_1=2 r_2$
$\therefore \quad \frac{l_1}{l_2}=\frac{2 r_2}{3\left(2 r_2\right)}$
$l_1: l_2=1: 3$
$n=\frac{1}{2 l} \sqrt{\frac{T}{m}}$
where, $m=$ mass per unit length of the string
$=\pi r^2 \rho$
$\therefore \quad n=\frac{1}{2 l} \sqrt{\frac{T}{\pi r^2 \rho}}$
For string $A$ first overtone,
$n_A=\frac{1}{l_1} \sqrt{\frac{T_1}{m_1}}=\frac{1}{l_1} \sqrt{\frac{T_1}{\pi r_1^2 \rho}}$
For string $B$ second overtone,
$n_B=\frac{3}{2 l_2} \sqrt{\frac{T_2}{\pi r_2^2 \rho_2}}$
According to question,
$n_A=n_B$
$\frac{1}{l_1} \sqrt{\frac{T_1}{\pi r_1^2 l_1}}=\frac{3}{2 l_2} \sqrt{\frac{T_2}{\pi r_2^2 l_2}}$
but $\quad T_1=T_2$
$\therefore \quad \frac{2 l_2}{3 l_1}=\sqrt{\frac{\pi r_1^2 \rho}{\pi r_2^2 \rho}}$
$\frac{4 l_2^2}{9 l_1^2}=\frac{r_1^2}{r_2^2}$
$\left(\frac{l_1}{l_2}\right)^2=\frac{4 r_2^2}{9 r_1^2}$
$\frac{2 l_2}{3 l_1}=\frac{r_1}{r_2}$
$\frac{l_2}{l_1}=\frac{3 r_1}{2 r_2}$
$\frac{l_1}{l_2}=\frac{2 r_2}{3 r_1}$
but $\quad r_1=2 r_2$
$\therefore \quad \frac{l_1}{l_2}=\frac{2 r_2}{3\left(2 r_2\right)}$
$l_1: l_2=1: 3$
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