• # question_answer Two uniform strings A and B made of steel an made to vibrate under the same tension. If the firs overtone of A is equal to the second overtone of I. and if the radius of A is twice that of B, the ratio of the lengths of the strings is: A)  2:1                             B)  3:2C)  3:4                             D) 1:3

Correct Answer: D

Solution :

According to question$2{{n}_{1}}=3{{n}_{2}}$ or $\frac{2}{2{{\ell }_{1}}}\sqrt{\frac{T}{{{m}_{1}}}}=\frac{3}{2{{\ell }_{1}}}\sqrt{\frac{T}{{{m}_{2}}}}$ or $\frac{{{\ell }_{1}}}{{{\ell }_{2}}}=\frac{2}{3}\sqrt{\frac{{{m}_{2}}}{{{m}_{1}}}}=\frac{2}{3}\sqrt{\frac{{{V}_{2}}\rho {{\ell }_{2}}}{{{V}_{1}}\rho {{\ell }_{1}}}}$ or $\frac{{{\ell }_{1}}}{{{\ell }_{2}}}=\frac{2}{3}\sqrt{\frac{r_{1}^{2}}{r_{1}^{2}}}=\frac{2}{3}\sqrt{{{\left( \frac{1}{2} \right)}^{2}}}$or $\frac{{{\ell }_{1}}}{{{\ell }_{2}}}=\frac{1}{3}$ Hence, the correction option is [d].

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