Question

Two strings $A$ and $B$ made of same material are stretched by same tension. The radius of cross-section of string $A$ is double the radius of cross-section of string $B$. A transverse wave travels through string $A$ with speed $v_A$ and through string $B$ with speed $v_B$. The ratio $\frac{v_A}{v_B}$ is

• A. $\frac{1}{2}$
• B. $2$
• C. $\frac{1}{4}$
• D. $4$

Answer 1

The correct option is A $\frac{1}{2}$

Velocity of a transverse wave in a stretched string is,
$v=\sqrt{\frac{T}{\mu }}$
where $T$ is the tension in the string and $\mu$ is mass per unit length of the string
$\mu =\frac{m}{l}=\frac{A\times l\times \rho }{l}$
$\mu =A\rho =\pi r^2\times \rho$
where $r$ is the radius of cross-section of the string and $\rho$ is the density of the material of the string.
$\Rightarrow v=\frac{1}{r}\sqrt{\frac{T}{\pi \rho }}$
Since $T,\rho$ are constant
$\Rightarrow v\propto \frac{1}{r}$
$\Rightarrow \frac{v_A}{v_B}=\frac{r_B}{r_A}=\left(\frac{r_B}{2r_B}\right)$
$\therefore \frac{v_A}{v_B}=\frac{1}{2}$