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JEE Main & Advanced Physics Wave Mechanics Question Bank Vibration of String

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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 [EAMCET 2003]

    A)             1: 2                                          

    B)            1 : 3

    C)            1 : 4                                          

    D)            1 : 6

    Correct Answer: B

    Solution :

                         First overtone of string\[A\]= Second overtone of string B.                    \[\Rightarrow \] Second harmonic of \[A\] = Third harmonic of B                    \[\Rightarrow \] \[{{n}_{2}}={{n}_{3}}\]\[\Rightarrow \] \[{{\left[ 2({{n}_{1}}) \right]}_{A}}={{\left[ 3({{n}_{1}}) \right]}_{B}}\] (\[\because \] \[{{n}_{1}}=\frac{1}{2l}\sqrt{\frac{T}{\pi {{r}^{2}}\rho }}\])                    Þ\[2\,\left[ \frac{1}{2{{l}_{A}}{{r}_{A}}}\sqrt{\frac{T}{\pi \rho }} \right]=3\,\left[ \frac{1}{2{{l}_{B}}{{r}_{B}}}\sqrt{\frac{T}{\pi \rho }} \right]\]                    \[\frac{{{l}_{A}}}{{{l}_{B}}}=\frac{2}{3}\frac{{{r}_{B}}}{{{r}_{A}}}\Rightarrow \frac{{{l}_{A}}}{{{l}_{B}}}=\frac{2}{3}\times \frac{{{r}_{B}}}{(2{{r}_{B}})}=\frac{1}{3}\]


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