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 length of the strings is:

• A. $1:2$
• B. $1:3$
• C. $1:4$
• D. $1:5$

Answer 1

The correct option is B $1:3$

$f=\frac{1}{2l}\sqrt{\frac{T}{\mu }}=\frac{1}{2l}\sqrt{\frac{T}{\rho \left(A\right)}}$
first overtone is $2f$
second overtone is $3f$
$3f_B=2f_A$
$3\times \frac{1}{2l_B}\times \frac{1}{\sqrt{(r_B{)}^2}}=1\times \frac{1}{2l_A}\times \frac{1}{\sqrt{(r_A{)}^2}}$
$\Rightarrow \frac{l_A}{l_B}=\frac{2}{3}\left(\frac{r_B}{r_A}\right)=\frac{2}{3}\times \frac{1}{2}=\frac{1}{3}$