Question

Two uniform stretched strings A and B, made of steel, are vibrating 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:

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

Answer 1

Answer: (C)

$1:3$

Given: $1^{st}$ overtone of $A=2^{nd}$ ovetone of $B$

$\Rightarrow \frac{1}{l_1}\sqrt{\frac{T}{\pi r_1^2}}=\frac{3}{2l_2}\sqrt{\frac{T}{\pi r_2^2}}$

$\Rightarrow \frac{r_2}{r_1}=\frac{3}{2}\left(\frac{l_1}{l_2}\right)$

Given that: $\frac{r_1}{r_2}=2$

$\Rightarrow \frac{1}{2}=$$\frac{3}{2}\left(\frac{l_1}{l_2}\right)$

$\Rightarrow l_1:l_2=1:3$